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Mathematics · Textbook solutions

Differentiation

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 363 questions

1.1 Derivatives of Composite Functions

72 q

Solved Examples

Worked · 20
  1. Differentiate the following w. r. t. xw.\ r.\ t.\ x.
    1.1.3 SolvedEx.1 (i)
    y=x2+5y = \sqrt{x^2 + 5}
  2. 1.1.3 SolvedEx.1 (ii)
    y=sin⁡(log⁡x)y = \sin(\log x)
  3. 1.1.3 SolvedEx.1 (iii)
    y=etan⁡xy = e^{\tan x}
  4. 1.1.3 SolvedEx.1 (iv)
    log⁡(x5+4)\log(x^5 + 4)
  5. 1.1.3 SolvedEx.1 (v)
    53cos⁡x−25^{3\cos x - 2}
  6. 1.1.3 SolvedEx.1 (vi)
    y=3(2x2−7)5y = \frac{3}{(2x^2 - 7)^5}
  7. Differentiate the following w. r. t. xw.\ r.\ t.\ x.
    1.1.3 SolvedEx.2 (i)
    y=sin⁡x3y = \sqrt{\sin x^3}
  8. 1.1.3 SolvedEx.2 (ii)
    y=cot⁡2(x3)y = \cot^2(x^3)
  9. 1.1.3 SolvedEx.2 (iii)
    y=log⁡[cos⁡(x5)]y = \log\left[\cos(x^5)\right]
  10. 1.1.3 SolvedEx.2 (iv)
    y=(x3+2x−3)4(x+cos⁡x)3y = (x^3 + 2x - 3)^4 (x + \cos x)^3
  11. 1.1.3 SolvedEx.2 (v)
    y=(1+cos⁡2x)4×x+tan⁡xy = (1 + \cos^2 x)^4 \times \sqrt{x + \sqrt{\tan x}}
  12. Differentiate the following w. r. t. xw.\ r.\ t.\ x.
    1.1.3 SolvedEx.3 (i)
    y=log⁡3(log⁡5x)y = \log_3(\log_5 x)
  13. 1.1.3 SolvedEx.3 (ii)
    y=log⁡[e3x⋅(3x−4)232x+53]y = \log\left[e^{3x} \cdot \frac{(3x - 4)^{\frac{2}{3}}}{\sqrt[3]{2x + 5}}\right]
  14. 1.1.3 SolvedEx.3 (iii)
    y=log⁡[1−cos⁡(3x2)1+cos⁡(3x2)]y = \log\left[\sqrt{\frac{1 - \cos\left(\frac{3x}{2}\right)}{1 + \cos\left(\frac{3x}{2}\right)}}\right]
  15. 1.1.3 SolvedEx.3 (iv)
    y=log⁡[x+x2+a2x2+a2−x]y = \log\left[\frac{x + \sqrt{x^2 + a^2}}{\sqrt{x^2 + a^2} - x}\right]
  16. 1.1.3 SolvedEx.3 (v)
    y=(4)log⁡2(sin⁡x)+(9)log⁡3(cos⁡x)y = (4)^{\log_2(\sin x)} + (9)^{\log_3(\cos x)}
  17. 1.1.3 SolvedEx.3 (vi)
    y=aalog⁡a(cot⁡x)y = a^{a^{\log_a(\cot x)}}
  18. 1.1.3 SolvedEx.4
    If f(x)=7g(x)−3f(x) = \sqrt{7g(x) - 3}, g(3)=4g(3) = 4 and g′(3)=5g'(3) = 5, find f′(3)f'(3).
  19. 1.1.3 SolvedEx.5
    If F(x)=G{3G[5G(x)]}F(x) = G\{3G[5G(x)]\}, G(0)=0G(0) = 0 and G′(0)=3G'(0) = 3, find F′(0)F'(0).
  20. 1.1.3 SolvedEx.6
    Select the appropriate hint from the hint basket and fill in the blank spaces in the following paragraph. [Activity] "Let f(x)=sin⁡xf(x) = \sin x and g(x)=log⁡xg(x) = \log x then f[g(x)]=f[g(x)] =______ and g[f(x)]=g[f(x)] =______. Now f′(x)=f'(x) =______ and g′(x)=g'(x) =______. The derivative of f[g(x)]f[g(x)] w. r. t. xw.\ r.\ t.\ x in terms of ff and gg is ______. Therefore ddx[f[g(x)]]=\frac{d}{dx}\left[f[g(x)]\right] =______ and [ddx[f[g(x)]]]x=1=\left[\frac{d}{dx}[f[g(x)]]\right]_{x=1} =______. The derivative of g[f(x)]g[f(x)] w. r. t. xw.\ r.\ t.\ x in terms of ff and gg is ______. Therefore ddx[g[f(x)]]=\frac{d}{dx}\left[g[f(x)]\right] =______ and [ddx[g[f(x)]]]x=π3=\left[\frac{d}{dx}[g[f(x)]]\right]_{x=\frac{\pi}{3}} =______." Hint basket : {f′[g(x)]⋅g′(x), cos⁡(log⁡x)x, 1, g′[f(x)]⋅f′(x), cot⁡x, 3, sin⁡(log⁡x), log⁡(sin⁡x), cos⁡x, 1x}\left\{f'[g(x)] \cdot g'(x),\ \frac{\cos(\log x)}{x},\ 1,\ g'[f(x)] \cdot f'(x),\ \cot x,\ \sqrt{3},\ \sin(\log x),\ \log(\sin x),\ \cos x,\ \frac{1}{x}\right\}

Exercise 1.1

Practice · 52
  1. Differentiate w. r. t. xx.
    Ex 1.1 Q.1 (i)
    (x3−2x−1)5(x^3 - 2x - 1)^5
  2. Ex 1.1 Q.1 (ii)
    (2x32−3x43−5)52\left(2x^{\frac{3}{2}} - 3x^{\frac{4}{3}} - 5\right)^{\frac{5}{2}}
  3. Ex 1.1 Q.1 (iii)
    x2+4x−7\sqrt{x^2 + 4x - 7}
  4. Ex 1.1 Q.1 (iv)
    x2+x2+1\sqrt{x^2 + \sqrt{x^2 + 1}}
  5. Ex 1.1 Q.1 (v)
    83(2x2−7x−5)113\frac{8}{3\sqrt[3]{\left(2x^2 - 7x - 5\right)^{11}}}
  6. Ex 1.1 Q.1 (vi)
    (3x−5−13x−5)5\left(\sqrt{3x - 5} - \frac{1}{\sqrt{3x - 5}}\right)^5
  7. Differentiate the following w.r.t. xx
    Ex 1.1 Q.2 (i)
    cos⁡(x2+a2)\cos\left(x^2 + a^2\right)
  8. Ex 1.1 Q.2 (ii)
    e(3x+2)+5\sqrt{e^{(3x + 2)} + 5}
  9. Ex 1.1 Q.2 (iii)
    log⁡[tan⁡(x2)]\log\left[\tan\left(\frac{x}{2}\right)\right]
  10. Ex 1.1 Q.2 (iv)
    tan⁡x\sqrt{\tan \sqrt{x}}
  11. Ex 1.1 Q.2 (v)
    cot⁡3[log⁡(x3)]\cot^3\left[\log\left(x^3\right)\right]
  12. Ex 1.1 Q.2 (vi)
    5sin⁡3x+35^{\sin^3 x + 3}
  13. Ex 1.1 Q.2 (vii)
    cosec⁡(cos⁡x)\operatorname{cosec}\left(\sqrt{\cos x}\right)
  14. Ex 1.1 Q.2 (viii)
    log⁡[cos⁡(x3−5)]\log\left[\cos\left(x^3 - 5\right)\right]
  15. Ex 1.1 Q.2 (ix)
    e3sin⁡2x−2cos⁡2xe^{3\sin^2 x - 2\cos^2 x}
  16. Ex 1.1 Q.2 (x)
    cos⁡2[log⁡(x2+7)]\cos^2\left[\log\left(x^2 + 7\right)\right]
  17. Ex 1.1 Q.2 (xi)
    tan⁡[cos⁡(sin⁡x)]\tan\left[\cos\left(\sin x\right)\right]
  18. Ex 1.1 Q.2 (xii)
    sec⁡[tan⁡(x4+4)]\sec\left[\tan\left(x^4 + 4\right)\right]
  19. Ex 1.1 Q.2 (xiii)
    elog⁡[(log⁡x)2−log⁡x2]e^{\log\left[(\log x)^2 - \log x^2\right]}
  20. Ex 1.1 Q.2 (xiv)
    sin⁡sin⁡x\sin \sqrt{\sin \sqrt{x}}
  21. Ex 1.1 Q.2 (xv)
    log⁡[sec⁡(ex2)]\log\left[\sec\left(e^{x^2}\right)\right]
  22. Ex 1.1 Q.2 (xvi)
    log⁡e2(log⁡x)\log_{e^2}\left(\log x\right)
  23. Ex 1.1 Q.2 (xvii)
    [log⁡[log⁡(log⁡x)]]2\left[\log\left[\log(\log x)\right]\right]^2
  24. Ex 1.1 Q.2 (xviii)
    sin⁡2x2−cos⁡2x2\sin^2 x^2 - \cos^2 x^2
  25. Differentiate the following w.r.t. xx
    Ex 1.1 Q.3 (i)
    (x2+4x+1)3+(x3−5x−2)4\left(x^2 + 4x + 1\right)^3 + \left(x^3 - 5x - 2\right)^4
  26. Ex 1.1 Q.3 (ii)
    (1+4x)5(3+x−x2)8\left(1 + 4x\right)^5 \left(3 + x - x^2\right)^8
  27. Ex 1.1 Q.3 (iii)
    x7−3x\frac{x}{\sqrt{7 - 3x}}
  28. Ex 1.1 Q.3 (iv)
    (x3−5)5(x3+3)3\frac{\left(x^3 - 5\right)^5}{\left(x^3 + 3\right)^3}
  29. Ex 1.1 Q.3 (v)
    (1+sin⁡2x)2(1+cos⁡2x)3\left(1 + \sin^2 x\right)^2 \left(1 + \cos^2 x\right)^3
  30. Ex 1.1 Q.3 (vi)
    cos⁡x+cos⁡x\sqrt{\cos x} + \sqrt{\cos \sqrt{x}}
  31. Ex 1.1 Q.3 (vii)
    log⁡(sec⁡3x+tan⁡3x)\log\left(\sec 3x + \tan 3x\right)
  32. Ex 1.1 Q.3 (viii)
    1+sin⁡x∘1−sin⁡x∘\frac{1 + \sin x^\circ}{1 - \sin x^\circ}
  33. Ex 1.1 Q.3 (ix)
    cot⁡(log⁡x2)−log⁡(cot⁡x2)\cot\left(\frac{\log x}{2}\right) - \log\left(\frac{\cot x}{2}\right)
  34. Ex 1.1 Q.3 (x)
    e2x−e−2xe2x+e−2x\frac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}
  35. Ex 1.1 Q.3 (xi)
    ex+1ex−1\frac{e^{\sqrt{x}} + 1}{e^{\sqrt{x}} - 1}
  36. Ex 1.1 Q.3 (xii)
    log⁡[tan⁡3x⋅sin⁡4x⋅(x2+7)7]\log\left[\tan^3 x \cdot \sin^4 x \cdot \left(x^2 + 7\right)^7\right]
  37. Ex 1.1 Q.3 (xiii)
    log⁡(1−cos⁡3x1+cos⁡3x)\log\left(\sqrt{\frac{1 - \cos 3x}{1 + \cos 3x}}\right)
  38. Ex 1.1 Q.3 (xiv)
    log⁡(1+cos⁡(5x2)1−cos⁡(5x2))\log\left(\sqrt{\frac{1 + \cos\left(\frac{5x}{2}\right)}{1 - \cos\left(\frac{5x}{2}\right)}}\right)
  39. Ex 1.1 Q.3 (xv)
    log⁡(1−sin⁡x1+sin⁡x)\log\left(\sqrt{\frac{1 - \sin x}{1 + \sin x}}\right)
  40. Ex 1.1 Q.3 (xvi)
    log⁡[42x(x2+52x3−4)32]\log\left[4^{2x}\left(\frac{x^2 + 5}{\sqrt{2x^3 - 4}}\right)^{\frac{3}{2}}\right]
  41. Ex 1.1 Q.3 (xvii)
    log⁡[ex2(5−4x)327−6x3]\log\left[\frac{e^{x^2}\left(5 - 4x\right)^{\frac{3}{2}}}{\sqrt[3]{7 - 6x}}\right]
  42. Ex 1.1 Q.3 (xviii)
    log⁡(acos⁡x(x2−3)3log⁡x)\log\left(\frac{a^{\cos x}}{\left(x^2 - 3\right)^3 \log x}\right)
  43. Ex 1.1 Q.3 (xix)
    y=(25)log⁡5(sec⁡x)−(16)log⁡4(tan⁡x)y = (25)^{\log_5 (\sec x)} - (16)^{\log_4 (\tan x)}
  44. Ex 1.1 Q.3 (xx)
    (x2+2)4x2+5\frac{\left(x^2 + 2\right)^4}{\sqrt{x^2 + 5}}
  45. A table of values of ff, gg, f′f' and g′g' is given
    xxf(x)f(x)g(x)g(x)f′(x)f'(x)g′(x)g'(x)
    216−3-34
    4345−6-6
    652−4-47
    Ex 1.1 Q.4 (i)
    If r(x)=f[g(x)]r(x) = f[g(x)] find r′(2)r'(2).
  46. Ex 1.1 Q.4 (ii)
    If R(x)=g[3+f(x)]R(x) = g[3 + f(x)] find R′(4)R'(4).
  47. Ex 1.1 Q.4 (iii)
    If s(x)=f[9−f(x)]s(x) = f[9 - f(x)] find s′(4)s'(4).
  48. Ex 1.1 Q.4 (iv)
    If S(x)=g[g(x)]S(x) = g[g(x)] find S′(6)S'(6).
  49. Ex 1.1 Q.5
    Assume that f′(3)=−1f'(3) = -1, g′(2)=5g'(2) = 5, g(2)=3g(2) = 3 and y=f[g(x)]y = f[g(x)] then [dydx]x=2=?\left[\frac{dy}{dx}\right]_{x=2} = ?
  50. Ex 1.1 Q.6
    If h(x)=4f(x)+3g(x)h(x) = \sqrt{4f(x) + 3g(x)}, f(1)=4f(1) = 4, g(1)=3g(1) = 3, f′(1)=3f'(1) = 3, g′(1)=4g'(1) = 4 find h′(1)h'(1).
  51. Ex 1.1 Q.7
    Find the xx co-ordinates of all the points on the curve y=sin⁡2x−2sin⁡xy = \sin 2x - 2 \sin x, 0≤x<2π0 \leq x < 2\pi where dydx=0\frac{dy}{dx} = 0.
  52. Ex 1.1 Q.8
    Select the appropriate hint from the hint basket and fill up the blank spaces in the following paragraph. [Activity] "Let f(x)=x2+5f(x) = x^2 + 5 and g(x)=ex+3g(x) = e^x + 3 then f[g(x)]=f[g(x)] = _____ and g[f(x)]=g[f(x)] = _____. Now f′(x)=f'(x) = _____ and g′(x)=g'(x) = _____. The derivative of f[g(x)]f[g(x)] w. r. t. xx in terms of ff and gg is _____. Therefore ddx[f[g(x)]]=\frac{d}{dx}\left[f[g(x)]\right] = _____ and [ddx[f[g(x)]]]x=0=\left[\frac{d}{dx}\left[f[g(x)]\right]\right]_{x=0} = _____. The derivative of g[f(x)]g[f(x)] w. r. t. xx in terms of ff and gg is _____. Therefore ddx[g[f(x)]]=\frac{d}{dx}\left[g[f(x)]\right] = _____ and [ddx[g[f(x)]]]x=−1=\left[\frac{d}{dx}\left[g[f(x)]\right]\right]_{x=-1} = _____." Hint basket : {f′[g(x)]⋅g′(x), 2e2x+6ex, 8, g′[f(x)]⋅f′(x), 2xex2+5, −2e6, e2x+6ex+14, ex2+5+3, 2x, ex}\left\{ f'[g(x)] \cdot g'(x),\ 2e^{2x} + 6e^x,\ 8,\ g'[f(x)] \cdot f'(x),\ 2xe^{x^2 + 5},\ -2e^6,\ e^{2x} + 6e^x + 14,\ e^{x^2 + 5} + 3,\ 2x,\ e^x \right\}

1.2 Derivatives of Inverse Functions

6 q

Solved Examples

Worked · 6
  1. Find the derivative of the function y=f(x)y = f(x) using the derivative of the inverse function x=f−1(y)x = f^{-1}(y) in the following
    1.2.3 SolvedEx.1 (i)
    y=x+43y = \sqrt[3]{x + 4}
  2. 1.2.3 SolvedEx.1 (ii)
    y=1+xy = \sqrt{1 + \sqrt{x}}
  3. 1.2.3 SolvedEx.1 (iii)
    y=log⁡xy = \log x
  4. 1.2.3 SolvedEx.2
    Find the derivative of the inverse of function y=2x3−6xy = 2x^3 - 6x and calculate its value at x=−2x = -2.
  5. 1.2.3 SolvedEx.3
    Let ff and gg be the inverse functions of each other. The following table lists a few values of ff, gg and f′f'
    xxf(x)f(x)g(x)g(x)f′(x)f'(x)
    −4-4221113\frac{1}{3}
    11−4-4−2-244
    find g′(−4)g'(-4).
  6. 1.2.3 SolvedEx.4
    Let f(x)=x5+2x−3f(x) = x^5 + 2x - 3. Find (f−1)′(−3)\left(f^{-1}\right)'(-3).

1.2 Derivatives of Inverse Trigonometric Functions

97 q

Solved Examples

Worked · 28
  1. 1.2.6 SolvedEx.1
    Using derivative prove that sin⁡−1x+cos⁡−1x=π2\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}.
  2. Differentiate the following w. r. t. x.
    1.2.6 SolvedEx.2 (i)
    sin⁡−1(x3)\sin^{-1}(x^3)
  3. 1.2.6 SolvedEx.2 (ii)
    cos⁡−1(2x2−x)\cos^{-1}(2x^2-x)
  4. 1.2.6 SolvedEx.2 (iii)
    sin⁡−1(2x)\sin^{-1}(2^x)
  5. 1.2.6 SolvedEx.2 (iv)
    cot⁡−1(1x2)\cot^{-1}\left(\frac{1}{x^2}\right)
  6. 1.2.6 SolvedEx.2 (v)
    cos⁡−1(1+x2)\cos^{-1}\left(\sqrt{\frac{1+x}{2}}\right)
  7. 1.2.6 SolvedEx.2 (vi)
    sin⁡2(sin⁡−1(x2))\sin^2\left(\sin^{-1}(x^2)\right)
  8. Differentiate the following w. r. t. x.
    1.2.6 SolvedEx.3 (i)
    cos⁡−1(4cos⁡3x−3cos⁡x)\cos^{-1}(4\cos^3 x-3\cos x)
  9. 1.2.6 SolvedEx.3 (ii)
    cos⁡−1[sin⁡(4x)]\cos^{-1}\left[\sin(4^x)\right]
  10. 1.2.6 SolvedEx.3 (iii)
    sin⁡−1(1−cos⁡x2)\sin^{-1}\left(\sqrt{\frac{1-\cos x}{2}}\right)
  11. 1.2.6 SolvedEx.3 (iv)
    tan⁡−1(1−cos⁡3xsin⁡3x)\tan^{-1}\left(\frac{1-\cos 3x}{\sin 3x}\right)
  12. 1.2.6 SolvedEx.3 (v)
    cot⁡−1(cos⁡x1+sin⁡x)\cot^{-1}\left(\frac{\cos x}{1+\sin x}\right)
  13. Differentiate the following w. r. t. x.
    1.2.6 SolvedEx.4 (i)
    sin⁡−1(2cos⁡x+3sin⁡x13)\sin^{-1}\left(\frac{2\cos x+3\sin x}{\sqrt{13}}\right)
  14. 1.2.6 SolvedEx.4 (ii)
    cos⁡−1(3sin⁡x2+4cos⁡x25)\cos^{-1}\left(\frac{3\sin x^2+4\cos x^2}{5}\right)
  15. 1.2.6 SolvedEx.4 (iii)
    sin⁡−1(acos⁡x−bsin⁡xa2+b2)\sin^{-1}\left(\frac{a\cos x-b\sin x}{\sqrt{a^2+b^2}}\right)
  16. Differentiate the following w. r. t. x.
    1.2.6 SolvedEx.5 (i)
    sin⁡−1(2x1+x2)\sin^{-1}\left(\frac{2x}{1+x^2}\right)
  17. 1.2.6 SolvedEx.5 (ii)
    cos⁡−1(2x1−x2)\cos^{-1}\left(2x\sqrt{1-x^2}\right)
  18. 1.2.6 SolvedEx.5 (iii)
    cosec⁡−1(13x−4x3)\operatorname{cosec}^{-1}\left(\frac{1}{3x-4x^3}\right)
  19. 1.2.6 SolvedEx.5 (iv)
    tan⁡−1(2ex1−e2x)\tan^{-1}\left(\frac{2e^x}{1-e^{2x}}\right)
  20. 1.2.6 SolvedEx.5 (v)
    cos⁡−1(1−9x21+9x2)\cos^{-1}\left(\frac{1-9x^2}{1+9x^2}\right)
  21. 1.2.6 SolvedEx.5 (vi)
    cos⁡−1(2x−2−x2x+2−x)\cos^{-1}\left(\frac{2^x-2^{-x}}{2^x+2^{-x}}\right)
  22. 1.2.6 SolvedEx.5 (vii)
    tan⁡−1(3−x3+x)\tan^{-1}\left(\sqrt{\frac{3-x}{3+x}}\right)
  23. 1.2.6 SolvedEx.5 (viii)
    sin⁡−1(51−x2−12x13)\sin^{-1}\left(\frac{5\sqrt{1-x^2}-12x}{13}\right)
  24. 1.2.6 SolvedEx.5 (ix)
    sin⁡−1(2x+11+4x)\sin^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)
  25. Differentiate the following w. r. t. x.
    1.2.6 SolvedEx.6 (i)
    tan⁡−1(4x1+21x2)\tan^{-1}\left(\frac{4x}{1+21x^2}\right)
  26. 1.2.6 SolvedEx.6 (ii)
    tan⁡−1(7x1−12x2)\tan^{-1}\left(\frac{7x}{1-12x^2}\right)
  27. 1.2.6 SolvedEx.6 (iii)
    cot⁡−1(bsin⁡x−acos⁡xasin⁡x+bcos⁡x)\cot^{-1}\left(\frac{b\sin x-a\cos x}{a\sin x+b\cos x}\right)
  28. 1.2.6 SolvedEx.6 (iv)
    tan⁡−1(5x+13−x−6x2)\tan^{-1}\left(\frac{5x+1}{3-x-6x^2}\right)

Exercise 1.2

Practice · 69
  1. Find the derivative of the function y=f(x)y=f(x) using the derivative of the inverse function x=f−1(y)x=f^{-1}(y) in the following
    Ex 1.2 Q.1 (i)
    y=xy=\sqrt{x}
  2. Ex 1.2 Q.1 (ii)
    y=2−xy=\sqrt{2-\sqrt{x}}
  3. Ex 1.2 Q.1 (iii)
    y=x−23y=\sqrt[3]{x-2}
  4. Ex 1.2 Q.1 (iv)
    y=log⁡(2x−1)y=\log(2x-1)
  5. Ex 1.2 Q.1 (v)
    y=2x+3y=2x+3
  6. Ex 1.2 Q.1 (vi)
    y=ex−3y=e^x-3
  7. Ex 1.2 Q.1 (vii)
    y=e2x−3y=e^{2x-3}
  8. Ex 1.2 Q.1 (viii)
    y=log⁡2(x2)y=\log_2\left(\frac{x}{2}\right)
  9. Find the derivative of the inverse function of the following
    Ex 1.2 Q.2 (i)
    y=x2⋅exy=x^2\cdot e^x
  10. Ex 1.2 Q.2 (ii)
    y=xcos⁡xy=x\cos x
  11. Ex 1.2 Q.2 (iii)
    y=x⋅7xy=x\cdot 7^x
  12. Ex 1.2 Q.2 (iv)
    y=x2+log⁡xy=x^2+\log x
  13. Ex 1.2 Q.2 (v)
    y=xlog⁡xy=x\log x
  14. Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them.
    Ex 1.2 Q.3 (i)
    y=x5+2x3+3xy=x^5+2x^3+3x, at x=1x=1
  15. Ex 1.2 Q.3 (ii)
    y=ex+3x+2y=e^x+3x+2, at x=0x=0
  16. Ex 1.2 Q.3 (iii)
    y=3x2+2log⁡x3y=3x^2+2\log x^3, at x=1x=1
  17. Ex 1.2 Q.3 (iv)
    y=sin⁡(x−2)+x2y=\sin(x-2)+x^2, at x=2x=2
  18. Ex 1.2 Q.4
    If f(x)=x3+x−2f(x)=x^3+x-2, find (f−1)′(0)(f^{-1})'(0).
  19. Using derivative prove
    Ex 1.2 Q.5 (i)
    tan⁡−1x+cot⁡−1x=π2\tan^{-1}x+\cot^{-1}x=\frac{\pi}{2}
  20. Ex 1.2 Q.5 (ii)
    sec⁡−1x+cosec⁡−1x=π2\sec^{-1}x+\operatorname{cosec}^{-1}x=\frac{\pi}{2} ... [for ∣x∣≥1|x|\geq 1]
  21. Differentiate the following w. r. t. x.
    Ex 1.2 Q.6 (i)
    tan⁡−1(log⁡x)\tan^{-1}(\log x)
  22. Ex 1.2 Q.6 (ii)
    cosec⁡−1(e−x)\operatorname{cosec}^{-1}(e^{-x})
  23. Ex 1.2 Q.6 (iii)
    cot⁡−1(x3)\cot^{-1}(x^3)
  24. Ex 1.2 Q.6 (iv)
    cot⁡−1(4x)\cot^{-1}(4^x)
  25. Ex 1.2 Q.6 (v)
    tan⁡−1(x)\tan^{-1}(\sqrt{x})
  26. Ex 1.2 Q.6 (vi)
    sin⁡−1(1+x22)\sin^{-1}\left(\sqrt{\frac{1+x^2}{2}}\right)
  27. Ex 1.2 Q.6 (vii)
    cos⁡−1(1−x2)\cos^{-1}(1-x^2)
  28. Ex 1.2 Q.6 (viii)
    sin⁡−1(x32)\sin^{-1}\left(x^{\frac{3}{2}}\right)
  29. Ex 1.2 Q.6 (ix)
    cos⁡3[cos⁡−1(x3)]\cos^3\left[\cos^{-1}(x^3)\right]
  30. Ex 1.2 Q.6 (x)
    sin⁡4[sin⁡−1(x)]\sin^4\left[\sin^{-1}(\sqrt{x})\right]
  31. Differentiate the following w. r. t. x.
    Ex 1.2 Q.7 (i)
    cot⁡−1[cot⁡(ex2)]\cot^{-1}\left[\cot\left(e^{x^2}\right)\right]
  32. Ex 1.2 Q.7 (ii)
    cosec⁡−1(1cos⁡(5x))\operatorname{cosec}^{-1}\left(\frac{1}{\cos(5^x)}\right)
  33. Ex 1.2 Q.7 (iii)
    cos⁡−1(1+cos⁡x2)\cos^{-1}\left(\sqrt{\frac{1+\cos x}{2}}\right)
  34. Ex 1.2 Q.7 (iv)
    cos⁡−1(1−cos⁡(x2)2)\cos^{-1}\left(\sqrt{\frac{1-\cos(x^2)}{2}}\right)
  35. Ex 1.2 Q.7 (v)
    tan⁡−1(1−tan⁡(x2)1+tan⁡(x2))\tan^{-1}\left(\frac{1-\tan\left(\frac{x}{2}\right)}{1+\tan\left(\frac{x}{2}\right)}\right)
  36. Ex 1.2 Q.7 (vi)
    cosec⁡−1(14cos⁡32x−3cos⁡2x)\operatorname{cosec}^{-1}\left(\frac{1}{4\cos^3 2x-3\cos 2x}\right)
  37. Ex 1.2 Q.7 (vii)
    tan⁡−1(1+cos⁡(x3)sin⁡(x3))\tan^{-1}\left(\frac{1+\cos\left(\frac{x}{3}\right)}{\sin\left(\frac{x}{3}\right)}\right)
  38. Ex 1.2 Q.7 (viii)
    cot⁡−1(sin⁡3x1+cos⁡3x)\cot^{-1}\left(\frac{\sin 3x}{1+\cos 3x}\right)
  39. Ex 1.2 Q.7 (ix)
    tan⁡−1(cos⁡7x1+sin⁡7x)\tan^{-1}\left(\frac{\cos 7x}{1+\sin 7x}\right)
  40. Ex 1.2 Q.7 (x)
    tan⁡−1(1+cos⁡x1−cos⁡x)\tan^{-1}\left(\sqrt{\frac{1+\cos x}{1-\cos x}}\right)
  41. Ex 1.2 Q.7 (xi)
    tan⁡−1(cosec⁡x+cot⁡x)\tan^{-1}(\operatorname{cosec} x+\cot x)
  42. Ex 1.2 Q.7 (xii)
    cot⁡−1(1+sin⁡(4x3)+1−sin⁡(4x3)1+sin⁡(4x3)−1−sin⁡(4x3))\cot^{-1}\left(\frac{\sqrt{1+\sin\left(\frac{4x}{3}\right)}+\sqrt{1-\sin\left(\frac{4x}{3}\right)}}{\sqrt{1+\sin\left(\frac{4x}{3}\right)}-\sqrt{1-\sin\left(\frac{4x}{3}\right)}}\right)
  43. Differentiate the following w. r. t. x.
    Ex 1.2 Q.8 (i)
    sin⁡−1(4sin⁡x+5cos⁡x41)\sin^{-1}\left(\frac{4\sin x+5\cos x}{\sqrt{41}}\right)
  44. Ex 1.2 Q.8 (ii)
    cos⁡−1(3cos⁡x−sin⁡x2)\cos^{-1}\left(\frac{\sqrt{3}\cos x-\sin x}{2}\right)
  45. Ex 1.2 Q.8 (iii)
    sin⁡−1(cos⁡x+sin⁡x2)\sin^{-1}\left(\frac{\cos\sqrt{x}+\sin\sqrt{x}}{\sqrt{2}}\right)
  46. Ex 1.2 Q.8 (iv)
    cos⁡−1(3cos⁡3x−4sin⁡3x5)\cos^{-1}\left(\frac{3\cos 3x-4\sin 3x}{5}\right)
  47. Ex 1.2 Q.8 (v)
    cos⁡−1(3cos⁡(ex)+2sin⁡(ex)13)\cos^{-1}\left(\frac{3\cos(e^x)+2\sin(e^x)}{\sqrt{13}}\right)
  48. Ex 1.2 Q.8 (vi)
    cosec⁡−1(106sin⁡(2x)−8cos⁡(2x))\operatorname{cosec}^{-1}\left(\frac{10}{6\sin(2^x)-8\cos(2^x)}\right)
  49. Differentiate the following w. r. t. x.
    Ex 1.2 Q.9 (i)
    cos⁡−1(1−x21+x2)\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)
  50. Ex 1.2 Q.9 (ii)
    tan⁡−1(2x1−x2)\tan^{-1}\left(\frac{2x}{1-x^2}\right)
  51. Ex 1.2 Q.9 (iii)
    sin⁡−1(1−x21+x2)\sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)
  52. Ex 1.2 Q.9 (iv)
    sin⁡−1(2x1−x2)\sin^{-1}\left(2x\sqrt{1-x^2}\right)
  53. Ex 1.2 Q.9 (v)
    cos⁡−1(3x−4x3)\cos^{-1}(3x-4x^3)
  54. Ex 1.2 Q.9 (vi)
    cos⁡−1(ex−e−xex+e−x)\cos^{-1}\left(\frac{e^x-e^{-x}}{e^x+e^{-x}}\right)
  55. Ex 1.2 Q.9 (vii)
    cos⁡−1(1−9x1+9x)\cos^{-1}\left(\frac{1-9^x}{1+9^x}\right)
  56. Ex 1.2 Q.9 (viii)
    sin⁡−1(4x+121+24x)\sin^{-1}\left(\frac{4^{x+\frac{1}{2}}}{1+2^{4x}}\right)
  57. Ex 1.2 Q.9 (ix)
    sin⁡−1(1−25x21+25x2)\sin^{-1}\left(\frac{1-25x^2}{1+25x^2}\right)
  58. Ex 1.2 Q.9 (x)
    sin⁡−1(1−x31+x3)\sin^{-1}\left(\frac{1-x^3}{1+x^3}\right)
  59. Ex 1.2 Q.9 (xi)
    tan⁡−1(2x521−x5)\tan^{-1}\left(\frac{2x^{\frac{5}{2}}}{1-x^5}\right)
  60. Ex 1.2 Q.9 (xii)
    cot⁡−1(1−x1+x)\cot^{-1}\left(\frac{1-\sqrt{x}}{1+\sqrt{x}}\right)
  61. Differentiate the following w. r. t. x.
    Ex 1.2 Q.10 (i)
    tan⁡−1(8x1−15x2)\tan^{-1}\left(\frac{8x}{1-15x^2}\right)
  62. Ex 1.2 Q.10 (ii)
    cot⁡−1(1+35x22x)\cot^{-1}\left(\frac{1+35x^2}{2x}\right)
  63. Ex 1.2 Q.10 (iii)
    tan⁡−1(2x1+3x)\tan^{-1}\left(\frac{2\sqrt{x}}{1+3x}\right)
  64. Ex 1.2 Q.10 (iv)
    tan⁡−1(2x+21−3(4x))\tan^{-1}\left(\frac{2^{x+2}}{1-3(4^x)}\right)
  65. Ex 1.2 Q.10 (v)
    tan⁡−1(2x1+22x+1)\tan^{-1}\left(\frac{2^x}{1+2^{2x+1}}\right)
  66. Ex 1.2 Q.10 (vi)
    cot⁡−1(a2−6x25ax)\cot^{-1}\left(\frac{a^2-6x^2}{5ax}\right)
  67. Ex 1.2 Q.10 (vii)
    tan⁡−1(a+btan⁡xb−atan⁡x)\tan^{-1}\left(\frac{a+b\tan x}{b-a\tan x}\right)
  68. Ex 1.2 Q.10 (viii)
    tan⁡−1(5−x6x2−5x−3)\tan^{-1}\left(\frac{5-x}{6x^2-5x-3}\right)
  69. Ex 1.2 Q.10 (ix)
    cot⁡−1(4−x−2x23x+2)\cot^{-1}\left(\frac{4-x-2x^2}{3x+2}\right)

1.3 Logarithmic Differentiation

5 q

Solved Examples

Worked · 5
  1. Differentiate the following w. r. t. x.
    1.3.1 SolvedEx.1 (i)
    ((x2+3)2 (x3+5)23(2x2+1)3)\left(\frac{(x^2+3)^2\ \sqrt[3]{(x^3+5)^2}}{\sqrt{(2x^2+1)^3}}\right)
  2. 1.3.1 SolvedEx.1 (ii)
    ex2 (tan⁡x)x2(1+x2)32 cos⁡3x\frac{e^{x^2}\ (\tan x)^{\frac{x}{2}}}{(1+x^2)^{\frac{3}{2}}\ \cos^3 x}
  3. 1.3.1 SolvedEx.1 (iii)
    (x+1)32 (2x+3)52 (3x+4)23(x+1)^{\frac{3}{2}}\ (2x+3)^{\frac{5}{2}}\ (3x+4)^{\frac{2}{3}}
  4. 1.3.1 SolvedEx.1 (iv)
    xa+xx+axx^a + x^x + a^x
  5. 1.3.1 SolvedEx.1 (v)
    (sin⁡x)tan⁡x−xlog⁡x(\sin x)^{\tan x} - x^{\log x}

1.3 Derivatives of Implicit Functions

52 q

Solved Examples

Worked · 8
  1. Find dydx\frac{dy}{dx} if
    1.3.3 SolvedEx.1 (i)
    x5+xy3+x2y+y4=4x^5 + xy^3 + x^2y + y^4 = 4
  2. 1.3.3 SolvedEx.1 (ii)
    y3+cos⁡(xy)=x2−sin⁡(x+y)y^3 + \cos(xy) = x^2 - \sin(x + y)
  3. 1.3.3 SolvedEx.1 (iii)
    x2+exy=y2+log⁡(x+y)x^2 + e^{xy} = y^2 + \log(x + y)
  4. 1.3.3 SolvedEx.2
    Find xm⋅yn=(x+y)m+nx^m \cdot y^n = (x + y)^{m + n}, then prove that dydx=yx\frac{dy}{dx} = \frac{y}{x}.
  5. 1.3.3 SolvedEx.3
    If sin⁡(pxm−qympxm+qym)=r\sin\left(\frac{px^m - qy^m}{px^m + qy^m}\right) = r, then show that dydx=yx\frac{dy}{dx} = \frac{y}{x}, where rr is a constant.
  6. 1.3.3 SolvedEx.4
    If sec⁡−1(x3+y3x3−y3)=2a\sec^{-1}\left(\frac{x^3 + y^3}{x^3 - y^3}\right) = 2a, then show that dydx=x2tan⁡2ay2\frac{dy}{dx} = \frac{x^2\tan^2 a}{y^2}, where aa is a constant.
  7. 1.3.3 SolvedEx.5
    If y=tan⁡x+tan⁡x+tan⁡x+…∞y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + \ldots \infty}}}, then show that dydx=sec⁡2x2y−1\frac{dy}{dx} = \frac{\sec^2 x}{2y - 1}.
  8. 1.3.3 SolvedEx.6
    If 1−x2+1−y2=a(x−y)\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y), then show that dydx=1−y21−x2\frac{dy}{dx} = \sqrt{\frac{1 - y^2}{1 - x^2}}.

Exercise 1.3

Practice · 44
  1. Differentiate the following w. r. t. x
    Ex 1.3 Q.1 (i)
    (x+1)2(x+2)3 (x+3)4\frac{(x+1)^2}{(x+2)^3\,(x+3)^4}
  2. Ex 1.3 Q.1 (ii)
    4x−1(2x+3) (5−2x)23\sqrt[3]{\frac{4x-1}{(2x+3)\,(5-2x)^2}}
  3. Ex 1.3 Q.1 (iii)
    (x2+3)32⋅sin⁡32x⋅2x2(x^2+3)^{\frac{3}{2}}\cdot\sin^3 2x\cdot 2^{x^2}
  4. Ex 1.3 Q.1 (iv)
    (x2+2x+2)32(x+3)3 (cos⁡x)x\frac{(x^2+2x+2)^{\frac{3}{2}}}{(\sqrt{x}+3)^3\,(\cos x)^x}
  5. Ex 1.3 Q.1 (v)
    x5⋅tan⁡34xsin⁡23x\frac{x^5\cdot\tan^3 4x}{\sin^2 3x}
  6. Ex 1.3 Q.1 (vi)
    xtan⁡−1xx^{\tan^{-1} x}
  7. Ex 1.3 Q.1 (vii)
    (sin⁡x)x(\sin x)^x
  8. Ex 1.3 Q.1 (viii)
    sin⁡xx\sin x^x
  9. Differentiate the following w. r. t. x.
    Ex 1.3 Q.2 (i)
    xe+xx+ex+eex^e + x^x + e^x + e^e
  10. Ex 1.3 Q.2 (ii)
    xxx+exxx^{x^x} + e^{x^x}
  11. Ex 1.3 Q.2 (iii)
    (log⁡x)x−(cos⁡x)cot⁡x(\log x)^x - (\cos x)^{\cot x}
  12. Ex 1.3 Q.2 (iv)
    xex+(log⁡x)sin⁡xx^{e^x} + (\log x)^{\sin x}
  13. Ex 1.3 Q.2 (v)
    etan⁡x+(log⁡x)tan⁡xe^{\tan x} + (\log x)^{\tan x}
  14. Ex 1.3 Q.2 (vi)
    (sin⁡x)tan⁡x+(cos⁡x)cot⁡x(\sin x)^{\tan x} + (\cos x)^{\cot x}
  15. Ex 1.3 Q.2 (vii)
    10xx+xx10+x10x10^{x^x} + x^{x^{10}} + x^{10^x}
  16. Ex 1.3 Q.2 (viii)
    [(tan⁡x)tan⁡x]tan⁡x\left[(\tan x)^{\tan x}\right]^{\tan x} at x=π4x = \frac{\pi}{4}
  17. Find dydx\frac{dy}{dx} if
    Ex 1.3 Q.3 (i)
    x+y=a\sqrt{x} + \sqrt{y} = \sqrt{a}
  18. Ex 1.3 Q.3 (ii)
    xx+yy=aax\sqrt{x} + y\sqrt{y} = a\sqrt{a}
  19. Ex 1.3 Q.3 (iii)
    x+xy+y=1x + \sqrt{xy} + y = 1
  20. Ex 1.3 Q.3 (iv)
    x3+x2y+xy2+y3=81x^3 + x^2 y + xy^2 + y^3 = 81
  21. Ex 1.3 Q.3 (v)
    x2y2−tan⁡−1x2+y2=cot⁡−1x2+y2x^2 y^2 - \tan^{-1}\sqrt{x^2+y^2} = \cot^{-1}\sqrt{x^2+y^2}
  22. Ex 1.3 Q.3 (vi)
    xey+yex=1xe^y + ye^x = 1
  23. Ex 1.3 Q.3 (vii)
    ex+y=cos⁡(x−y)e^{x+y} = \cos(x-y)
  24. Ex 1.3 Q.3 (viii)
    cos⁡(xy)=x+y\cos(xy) = x + y
  25. Ex 1.3 Q.3 (ix)
    eex−y=xye^{e^{x-y}} = \frac{x}{y}
  26. Ex 1.3 Q.3 (x)
    x+sin⁡(x+y)=y−cos⁡(x−y)x + \sin(x+y) = y - \cos(x-y)
  27. Show that dydx=yx\frac{dy}{dx} = \frac{y}{x} in the following, where aa and pp are constants.
    Ex 1.3 Q.4 (i)
    x7y5=(x+y)12x^7 y^5 = (x+y)^{12}
  28. Ex 1.3 Q.4 (ii)
    xpy4=(x+y)p+4, p∈Nx^p y^4 = (x+y)^{p+4},\ p \in N
  29. Ex 1.3 Q.4 (iii)
    sec⁡(x5+y5x5−y5)=a2\sec\left(\frac{x^5+y^5}{x^5-y^5}\right) = a^2
  30. Ex 1.3 Q.4 (iv)
    tan⁡−1(3x2−4y23x2+4y2)=a2\tan^{-1}\left(\frac{3x^2-4y^2}{3x^2+4y^2}\right) = a^2
  31. Ex 1.3 Q.4 (v)
    cos⁡−1(7x4+5y47x4−5y4)=tan⁡−1a\cos^{-1}\left(\frac{7x^4+5y^4}{7x^4-5y^4}\right) = \tan^{-1} a
  32. Ex 1.3 Q.4 (vi)
    log⁡(x20−y20x20+y20)=20\log\left(\frac{x^{20}-y^{20}}{x^{20}+y^{20}}\right) = 20
  33. Ex 1.3 Q.4 (vii)
    ex7−y7x7+y7=ae^{\frac{x^7-y^7}{x^7+y^7}} = a
  34. Ex 1.3 Q.4 (viii)
    sin⁡(x3−y3x3+y3)=a3\sin\left(\frac{x^3-y^3}{x^3+y^3}\right) = a^3
  35. Ex 1.3 Q.5 (i)
    If log⁡(x+y)=log⁡(xy)+p\log(x+y) = \log(xy) + p, where pp is constant then prove that dydx=−y2x2\frac{dy}{dx} = -\frac{y^2}{x^2}.
  36. Ex 1.3 Q.5 (ii)
    If log⁡10(x3−y3x3+y3)=2\log_{10}\left(\frac{x^3-y^3}{x^3+y^3}\right) = 2, show that dydx=−99x2101y2\frac{dy}{dx} = -\frac{99x^2}{101y^2}.
  37. Ex 1.3 Q.5 (iii)
    If log⁡5(x4+y4x4−y4)=2\log_5\left(\frac{x^4+y^4}{x^4-y^4}\right) = 2, show that dydx=−12x313y3\frac{dy}{dx} = -\frac{12x^3}{13y^3}.
  38. Ex 1.3 Q.5 (iv)
    If ex+ey=ex+ye^x + e^y = e^{x+y}, then show that dydx=−ey−x\frac{dy}{dx} = -e^{y-x}.
  39. Ex 1.3 Q.5 (v)
    If sin⁡−1(x5−y5x5+y5)=π6\sin^{-1}\left(\frac{x^5-y^5}{x^5+y^5}\right) = \frac{\pi}{6}, show that dydx=x43y4\frac{dy}{dx} = \frac{x^4}{3y^4}.
  40. Ex 1.3 Q.5 (vi)
    If xy=ex−yx^y = e^{x-y}, then show that dydx=log⁡x(1+log⁡x)2\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}.
  41. Ex 1.3 Q.5 (vii)
    If y=cos⁡x+cos⁡x+cos⁡x+...∞y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + ...\infty}}}, then show that dydx=sin⁡x1−2y\frac{dy}{dx} = \frac{\sin x}{1-2y}.
  42. Ex 1.3 Q.5 (viii)
    If y=log⁡x+log⁡x+log⁡x+...∞y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ...\infty}}}, then show that dydx=1x(2y−1)\frac{dy}{dx} = \frac{1}{x(2y-1)}.
  43. Ex 1.3 Q.5 (ix)
    If y=xxx⋅⋅⋅∞y = x^{x^{x^{\cdot^{\cdot^{\cdot\infty}}}}}, then show that dydx=y2x(1−log⁡y)\frac{dy}{dx} = \frac{y^2}{x(1-\log y)}.
  44. Ex 1.3 Q.5 (x)
    If ey=yxe^y = y^x, then show that dydx=(log⁡y)2log⁡y−1\frac{dy}{dx} = \frac{(\log y)^2}{\log y - 1}.

1.4 Derivatives of Parametric Functions

11 q

Solved Examples

Worked · 11
  1. Find dydx\frac{dy}{dx} if
    1.4.2 SolvedEx.1 (i)
    x=at4x = at^4, y=2at2y = 2at^2
  2. 1.4.2 SolvedEx.1 (ii)
    x=t−tx = t - \sqrt{t}, y=t+ty = t + \sqrt{t}
  3. 1.4.2 SolvedEx.1 (iii)
    x=cos⁡(log⁡t)x = \cos(\log t), y=log⁡(cos⁡t)y = \log(\cos t)
  4. 1.4.2 SolvedEx.1 (iv)
    x=a(θ+sin⁡θ)x = a(\theta + \sin\theta), y=a(1−cos⁡θ)y = a(1 - \cos\theta)
  5. 1.4.2 SolvedEx.1 (v)
    x=1−t2x = \sqrt{1 - t^2}, y=sin⁡−1ty = \sin^{-1} t
  6. Find dydx\frac{dy}{dx} if
    1.4.2 SolvedEx.2 (i)
    x=sec⁡2θx = \sec^2\theta, y=tan⁡3θy = \tan^3\theta, at θ=π3\theta = \frac{\pi}{3}
  7. 1.4.2 SolvedEx.2 (ii)
    x=t+1tx = t + \frac{1}{t}, y=1t2y = \frac{1}{t^2}, at t=12t = \frac{1}{2}
  8. 1.4.2 SolvedEx.2 (iii)
    x=3cos⁡t−2cos⁡3tx = 3\cos t - 2\cos^3 t, y=3sin⁡t−2sin⁡3ty = 3\sin t - 2\sin^3 t, at t=π6t = \frac{\pi}{6}
  9. 1.4.2 SolvedEx.3
    If x2+y2=t+1tx^2 + y^2 = t + \frac{1}{t} and x4+y4=t2+1t2x^4 + y^4 = t^2 + \frac{1}{t^2}, then show that x3ydydx=−1x^3 y \frac{dy}{dx} = -1.
  10. 1.4.2 SolvedEx.4
    If x=a(t−1t)x = a\left(t - \frac{1}{t}\right) and y=b(t+1t)y = b\left(t + \frac{1}{t}\right), then show that dydx=b2xa2y\frac{dy}{dx} = \frac{b^2 x}{a^2 y}.
  11. 1.4.2 SolvedEx.5
    If x=asin⁡−1tx = \sqrt{a^{\sin^{-1} t}} and y=acos⁡−1ty = \sqrt{a^{\cos^{-1} t}}, then show that dydx=−yx\frac{dy}{dx} = -\frac{y}{x}.

1.4 Differentiation of One Function w.r.t. Another

32 q

Solved Examples

Worked · 3
  1. 1.4.3 SolvedEx.1
    Find the derivative of 7x7^x w.r.t. x7x^7.
  2. 1.4.3 SolvedEx.2
    Find the derivative of cos⁡−1x\cos^{-1} x w.r.t. 1−x2\sqrt{1-x^2}.
  3. 1.4.3 SolvedEx.3
    Find the derivative of tan⁡−1(1+x2−1x)\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) w.r.t. sin⁡−1(2x1+x2)\sin^{-1}\left(\frac{2x}{1+x^2}\right).

Exercise 1.4

Practice · 29
  1. Find dydx\frac{dy}{dx} if
    Ex 1.4 Q.1 (i)
    x=at2x = at^2, y=2aty = 2at
  2. Ex 1.4 Q.1 (ii)
    x=acot⁡θx = a \cot \theta, y=bcosec⁡θy = b \operatorname{cosec} \theta
  3. Ex 1.4 Q.1 (iii)
    x=a2+m2x = \sqrt{a^2 + m^2}, y=log⁡(a2+m2)y = \log (a^2 + m^2)
  4. Ex 1.4 Q.1 (iv)
    x=sin⁡θx = \sin \theta, y=tan⁡θy = \tan \theta
  5. Ex 1.4 Q.1 (v)
    x=a(1−cos⁡θ)x = a(1 - \cos \theta), y=b(θ−sin⁡θ)y = b(\theta - \sin \theta)
  6. Ex 1.4 Q.1 (vi)
    x=(t+1t)ax = \left(t + \frac{1}{t}\right)^a, y=at+1ty = a^{t + \frac{1}{t}}, where a>0a > 0, a≠1a \neq 1 and t≠0t \neq 0.
  7. Ex 1.4 Q.1 (vii)
    x=cos⁡−1(2t1+t2)x = \cos^{-1}\left(\frac{2t}{1+t^2}\right), y=sec⁡−1(1+t2)y = \sec^{-1}\left(\sqrt{1+t^2}\right)
  8. Ex 1.4 Q.1 (viii)
    x=cos⁡−1(4t3−3t)x = \cos^{-1}(4t^3 - 3t), y=tan⁡−1(1−t2t)y = \tan^{-1}\left(\frac{\sqrt{1-t^2}}{t}\right)
  9. Find dydx\frac{dy}{dx} if
    Ex 1.4 Q.2 (i)
    x=cosec⁡2θx = \operatorname{cosec}^2 \theta, y=cot⁡3θy = \cot^3 \theta, at θ=π6\theta = \frac{\pi}{6}
  10. Ex 1.4 Q.2 (ii)
    x=acos⁡3θx = a \cos^3 \theta, y=asin⁡3θy = a \sin^3 \theta, at θ=π3\theta = \frac{\pi}{3}
  11. Ex 1.4 Q.2 (iii)
    x=t2+t+1x = t^2 + t + 1, y=sin⁡(πt2)+cos⁡(πt2)y = \sin\left(\frac{\pi t}{2}\right) + \cos\left(\frac{\pi t}{2}\right), at t=1t = 1
  12. Ex 1.4 Q.2 (iv)
    x=2cos⁡t+cos⁡2tx = 2 \cos t + \cos 2t, y=2sin⁡t−sin⁡2ty = 2 \sin t - \sin 2t, at t=π4t = \frac{\pi}{4}
  13. Ex 1.4 Q.2 (v)
    x=t+2sin⁡(πt)x = t + 2 \sin(\pi t), y=3t−cos⁡(πt)y = 3t - \cos(\pi t), at t=12t = \frac{1}{2}
  14. Ex 1.4 Q.3 (i)
    If x=asec⁡θ−tan⁡θx = a\sqrt{\sec \theta - \tan \theta}, y=asec⁡θ+tan⁡θy = a\sqrt{\sec \theta + \tan \theta}, then show that dydx=−yx\frac{dy}{dx} = -\frac{y}{x}.
  15. Ex 1.4 Q.3 (ii)
    If x=esin⁡3tx = e^{\sin 3t}, y=ecos⁡3ty = e^{\cos 3t}, then show that dydx=−ylog⁡xxlog⁡y\frac{dy}{dx} = -\frac{y \log x}{x \log y}.
  16. Ex 1.4 Q.3 (iii)
    If x=t+1t−1x = \frac{t+1}{t-1}, y=t−1t+1y = \frac{t-1}{t+1}, then show that y2+dydx=0y^2 + \frac{dy}{dx} = 0.
  17. Ex 1.4 Q.3 (iv)
    If x=acos⁡3tx = a \cos^3 t, y=asin⁡3ty = a \sin^3 t, then show that dydx=−(yx)13\frac{dy}{dx} = -\left(\frac{y}{x}\right)^{\frac{1}{3}}.
  18. Ex 1.4 Q.3 (v)
    If x=2cos⁡4(t+3)x = 2 \cos^4 (t+3), y=3sin⁡4(t+3)y = 3 \sin^4 (t+3), show that dydx=−3y2x\frac{dy}{dx} = -\sqrt{\frac{3y}{2x}}.
  19. Ex 1.4 Q.3 (vi)
    If x=log⁡(1+t2)x = \log (1 + t^2), y=t−tan⁡−1ty = t - \tan^{-1} t, show that dydx=ex−12\frac{dy}{dx} = \frac{\sqrt{e^x - 1}}{2}.
  20. Ex 1.4 Q.3 (vii)
    If x=sin⁡−1(et)x = \sin^{-1} (e^t), y=1−e2ty = \sqrt{1 - e^{2t}}, show that sin⁡x+dydx=0\sin x + \frac{dy}{dx} = 0.
  21. Ex 1.4 Q.3 (viii)
    If x=2bt1+t2x = \frac{2bt}{1+t^2}, y=a(1−t21+t2)y = a\left(\frac{1-t^2}{1+t^2}\right), show that dxdy=−b2ya2x\frac{dx}{dy} = -\frac{b^2 y}{a^2 x}.
  22. Ex 1.4 Q.4 (i)
    Differentiate xsin⁡xx \sin x w.r.t. tan⁡x\tan x.
  23. Ex 1.4 Q.4 (ii)
    Differentiate sin⁡−1(2x1+x2)\sin^{-1}\left(\frac{2x}{1+x^2}\right) w.r.t. cos⁡−1(1−x21+x2)\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right).
  24. Ex 1.4 Q.4 (iii)
    Differentiate tan⁡−1(x1−x2)\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right) w.r.t. sec⁡−1(12x2−1)\sec^{-1}\left(\frac{1}{2x^2-1}\right).
  25. Ex 1.4 Q.4 (iv)
    Differentiate cos⁡−1(1−x21+x2)\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) w.r.t. tan⁡−1x\tan^{-1} x.
  26. Ex 1.4 Q.4 (v)
    Differentiate 3x3^x w.r.t. log⁡x3\log_x 3.
  27. Ex 1.4 Q.4 (vi)
    Differentiate tan⁡−1(cos⁡x1+sin⁡x)\tan^{-1}\left(\frac{\cos x}{1+\sin x}\right) w.r.t. sec⁡−1x\sec^{-1} x.
  28. Ex 1.4 Q.4 (vii)
    Differentiate xxx^x w.r.t. xsin⁡xx^{\sin x}.
  29. Ex 1.4 Q.4 (viii)
    Differentiate tan⁡−1(1+x2−1x)\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) w.r.t. tan⁡−1(2x1−x21−2x2)\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right).

1.5 Higher Order Derivatives

10 q

Solved Examples

Worked · 10
  1. Find the second order derivative of the following :
    1.5.1 SolvedEx.1 (i)
    x3+7x2−2x−9x^3 + 7x^2 - 2x - 9
  2. 1.5.1 SolvedEx.1 (ii)
    x2exx^2 e^x
  3. 1.5.1 SolvedEx.1 (iii)
    e2xsin⁡3xe^{2x} \sin 3x
  4. 1.5.1 SolvedEx.1 (iv)
    x2log⁡xx^2 \log x
  5. 1.5.1 SolvedEx.1 (v)
    sin⁡(log⁡x)\sin (\log x)
  6. Find d2ydx2\frac{d^2y}{dx^2} if,
    1.5.1 SolvedEx.2 (i)
    x=cot⁡−1(1−t2t)x = \cot^{-1}\left(\frac{\sqrt{1 - t^2}}{t}\right) and x=cosec⁡−1(1+t22t)x = \operatorname{cosec}^{-1}\left(\frac{1 + t^2}{2t}\right)
  7. 1.5.1 SolvedEx.2 (ii)
    x=acos⁡3θx = a \cos^3 \theta, y=bsin⁡3θy = b \sin^3 \theta at θ=π4\theta = \frac{\pi}{4}
  8. 1.5.1 SolvedEx.3
    If ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 then show that d2ydx2=0\frac{d^2y}{dx^2} = 0.
  9. 1.5.1 SolvedEx.4
    If y=cos⁡(mcos⁡−1x)y = \cos (m \cos^{-1} x) then show that (1−x2)d2ydx2−xdydx+m2y=0(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2 y = 0.
  10. 1.5.1 SolvedEx.5
    If x=sin⁡tx = \sin t, y=emty = e^{mt} then show that (1−x2)d2ydx2−xdydx−m2y=0(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} - m^2 y = 0.

1.5 Successive Differentiation (nth Order Derivative)

40 q

Solved Examples

Worked · 6
  1. Find the nthn^{th} derivative of the following :
    1.5.2 SolvedEx.1 (i)
    xmx^m
  2. 1.5.2 SolvedEx.1 (ii)
    1ax+b\frac{1}{ax + b}
  3. 1.5.2 SolvedEx.1 (iii)
    log⁡x\log x
  4. 1.5.2 SolvedEx.1 (iv)
    sin⁡x\sin x
  5. 1.5.2 SolvedEx.1 (v)
    cos⁡(ax+b)\cos(ax + b)
  6. 1.5.2 SolvedEx.1 (vi)
    eaxsin⁡(bx+c)e^{ax}\sin(bx + c)

Exercise 1.5

Practice · 34
  1. Find the second order derivative of the following :
    Ex 1.5 Q.1 (i)
    2x5−4x3−2x2−92x^5 - 4x^3 - \frac{2}{x^2} - 9
  2. Ex 1.5 Q.1 (ii)
    e2x⋅tan⁡xe^{2x} \cdot \tan x
  3. Ex 1.5 Q.1 (iii)
    e4x⋅cos⁡5xe^{4x} \cdot \cos 5x
  4. Ex 1.5 Q.1 (iv)
    x3log⁡xx^3 \log x
  5. Ex 1.5 Q.1 (v)
    log⁡(log⁡x)\log (\log x)
  6. Ex 1.5 Q.1 (vi)
    xxx^x
  7. Find d2ydx2\frac{d^2y}{dx^2} of the following :
    Ex 1.5 Q.2 (i)
    x=a (θ−sin⁡θ)x = a\,(\theta - \sin \theta), y=a (1−cos⁡θ)y = a\,(1 - \cos \theta)
  8. Ex 1.5 Q.2 (ii)
    x=2at2x = 2at^2, y=4aty = 4at
  9. Ex 1.5 Q.2 (iii)
    x=sin⁡θx = \sin \theta, y=sin⁡3θy = \sin^3 \theta when θ=π2\theta = \frac{\pi}{2}
  10. Ex 1.5 Q.2 (iv)
    x=acos⁡θx = a \cos \theta, y=bsin⁡θy = b \sin \theta at θ=π4\theta = \frac{\pi}{4}
  11. Ex 1.5 Q.3 (i)
    If x=at2x = at^2 and y=2aty = 2at then show that xyd2ydx2+a=0xy \frac{d^2y}{dx^2} + a = 0
  12. Ex 1.5 Q.3 (ii)
    If y=emtan⁡−1xy = e^{m \tan^{-1} x}, show that (1+x2)d2ydx2+(2x−m)dydx=0(1 + x^2) \frac{d^2y}{dx^2} + (2x - m) \frac{dy}{dx} = 0
  13. Ex 1.5 Q.3 (iii)
    If x=cos⁡tx = \cos t, y=emty = e^{mt} show that (1−x2)d2ydx2−xdydx−m2y=0(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} - m^2 y = 0
  14. Ex 1.5 Q.3 (iv)
    If y=x+tan⁡xy = x + \tan x, show that cos⁡2x⋅d2ydx2−2y+2x=0\cos^2 x \cdot \frac{d^2y}{dx^2} - 2y + 2x = 0
  15. Ex 1.5 Q.3 (v)
    If y=eax⋅sin⁡(bx)y = e^{ax} \cdot \sin (bx), show that y2−2ay1+(a2+b2) y=0y_2 - 2a y_1 + (a^2 + b^2)\, y = 0
  16. Ex 1.5 Q.3 (vi)
    If sec⁡−1(7x3−5y37x3+5y3)=m\sec^{-1}\left(\frac{7x^3 - 5y^3}{7x^3 + 5y^3}\right) = m, show that d2ydx2=0\frac{d^2y}{dx^2} = 0.
  17. Ex 1.5 Q.3 (vii)
    If 2y=x+1+x−12y = \sqrt{x + 1} + \sqrt{x - 1}, show that 4(x2−1) y2+4x y1−y=04(x^2 - 1)\, y_2 + 4x\, y_1 - y = 0.
  18. Ex 1.5 Q.3 (viii)
    If y=log⁡(x+x2+a2)my = \log \left(x + \sqrt{x^2 + a^2}\right)^m, show that (x2+a2)d2ydx2+xdydx=0(x^2 + a^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = 0
  19. Ex 1.5 Q.3 (ix)
    If y=sin⁡(mcos⁡−1x)y = \sin (m \cos^{-1} x) then show that (1−x2)d2ydx2−xdydx+m2y=0(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2 y = 0
  20. Ex 1.5 Q.3 (x)
    If y=log⁡(log⁡2x)y = \log (\log 2x), show that x y2+y1(1+x y1)=0x\, y_2 + y_1 (1 + x\, y_1) = 0.
  21. Ex 1.5 Q.3 (xi)
    If x2+6xy+y2=10x^2 + 6xy + y^2 = 10, show that d2ydx2=80(3x+y)3\frac{d^2y}{dx^2} = \frac{80}{(3x + y)^3}.
  22. Ex 1.5 Q.3 (xii)
    If x=asin⁡t−bcos⁡tx = a \sin t - b \cos t, y=acos⁡t+bsin⁡ty = a \cos t + b \sin t, show that d2ydx2=−x2+y2y3\frac{d^2y}{dx^2} = -\frac{x^2 + y^2}{y^3}.
  23. Find the nthn^{\text{th}} derivative of the following :
    Ex 1.5 Q.4 (i)
    (ax+b)m(ax + b)^m
  24. Ex 1.5 Q.4 (ii)
    1x\frac{1}{x}
  25. Ex 1.5 Q.4 (iii)
    eax+be^{ax + b}
  26. Ex 1.5 Q.4 (iv)
    apx+qa^{px + q}
  27. Ex 1.5 Q.4 (v)
    log⁡(ax+b)\log (ax + b)
  28. Ex 1.5 Q.4 (vi)
    cos⁡x\cos x
  29. Ex 1.5 Q.4 (vii)
    sin⁡(ax+b)\sin (ax + b)
  30. Ex 1.5 Q.4 (viii)
    cos⁡(3−2x)\cos (3 - 2x)
  31. Ex 1.5 Q.4 (ix)
    log⁡(2x+3)\log (2x + 3)
  32. Ex 1.5 Q.4 (x)
    13x−5\frac{1}{3x - 5}
  33. Ex 1.5 Q.4 (xi)
    y=eax⋅cos⁡(bx+c)y = e^{ax} \cdot \cos (bx + c)
  34. Ex 1.5 Q.4 (xii)
    y=e8x⋅cos⁡(6x+7)y = e^{8x} \cdot \cos (6x + 7)

Miscellaneous Exercise 1

38 q

Choose the correct option

Practice · 12
  1. Misc I Q.1
    Let f(1)=3f(1) = 3, f′(1)=−13f'(1) = -\frac{1}{3}, g(1)=−4g(1) = -4 and g′(1)=−83g'(1) = -\frac{8}{3}. The derivative of [f(x)]2+[g(x)]2\sqrt{[f(x)]^2 + [g(x)]^2} w. r. t. xx at x=1x = 1 is

    Tap an option to check your answer.

  2. Misc I Q.2
    If y=sec⁡(tan⁡−1x)y = \sec(\tan^{-1} x) then dydx\frac{dy}{dx} at x=1x = 1, is equal to :

    Tap an option to check your answer.

  3. Misc I Q.3
    If f(x)=sin⁡−1(4x+121+24x)f(x) = \sin^{-1}\left(\frac{4^{x + \frac{1}{2}}}{1 + 2^{4x}}\right), which of the following is not the derivative of f(x)f(x)

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  4. Misc I Q.4
    If xy=yxx^y = y^x, then dydx=\frac{dy}{dx} = ...

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  5. Misc I Q.5
    If y=sin⁡(2sin⁡−1x)y = \sin(2 \sin^{-1} x), then dydx=\frac{dy}{dx} = ...

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  6. Misc I Q.6
    If y=tan⁡−1(x1+1−x2)+sin⁡[2tan⁡−1(1−x1+x)]y = \tan^{-1}\left(\frac{x}{1 + \sqrt{1 - x^2}}\right) + \sin\left[2 \tan^{-1}\left(\sqrt{\frac{1 - x}{1 + x}}\right)\right], then dydx=\frac{dy}{dx} = ...

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  7. Misc I Q.7
    If yy is a function of xx and log⁡(x+y)=2xy\log(x + y) = 2xy, then the value of y′(0)=y'(0) = ...

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  8. Misc I Q.8
    If gg is the inverse of a function ff and f′(x)=11+x7f'(x) = \frac{1}{1 + x^7}, then the value of g′(x)g'(x) is equal to :

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  9. Misc I Q.9
    If xy+1+yx+1=0x\sqrt{y + 1} + y\sqrt{x + 1} = 0 and x≠yx \neq y then dydx=\frac{dy}{dx} = ...

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  10. Misc I Q.10
    If y=tan⁡−1(a−xa+x)y = \tan^{-1}\left(\sqrt{\frac{a - x}{a + x}}\right), where −a<x<a-a < x < a then dydx=\frac{dy}{dx} = ...

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  11. Misc I Q.11
    If x=a(cos⁡θ+θsin⁡θ)x = a(\cos\theta + \theta\sin\theta), y=a(sin⁡θ−θcos⁡θ)y = a(\sin\theta - \theta\cos\theta) then [d2ydx2]θ=π4=\left[\frac{d^2y}{dx^2}\right]_{\theta = \frac{\pi}{4}} = ...

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  12. Misc I Q.12
    If y=acos⁡(log⁡x)y = a\cos(\log x) and Ad2ydx2+Bdydx+C=0A\frac{d^2y}{dx^2} + B\frac{dy}{dx} + C = 0, then the values of AA, BB, CC are ...

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Solve the following

Practice · 26
  1. Misc II Q.1
    f(x)={−x,for −2≤x<02x,for 0≤x≤218−x4,for 2<x≤7f(x) = \begin{cases} -x, & \text{for } -2 \le x < 0 \\ 2x, & \text{for } 0 \le x \le 2 \\ \frac{18 - x}{4}, & \text{for } 2 < x \le 7 \end{cases} g(x)={6−3x,for 0≤x≤22x−43,for 2<x≤7g(x) = \begin{cases} 6 - 3x, & \text{for } 0 \le x \le 2 \\ \frac{2x - 4}{3}, & \text{for } 2 < x \le 7 \end{cases} Let u(x)=f[g(x)]u(x) = f[g(x)], v(x)=g[f(x)]v(x) = g[f(x)] and w(x)=g[g(x)]w(x) = g[g(x)]. Find each derivative at x=1x = 1, if it exists i.e. find u′(1)u'(1), v′(1)v'(1) and w′(1)w'(1). if it doesn't exist then explain why ?
  2. Misc II Q.2
    The values of f(x)f(x), g(x)g(x), f′(x)f'(x) and g′(x)g'(x) are given in the following table.
    xxf(x)f(x)g(x)g(x)f′(x)f'(x)g′(x)g'(x)
    −1-13322−3-344
    2222−1-1−5-5−4-4
    Match the following.
    A Group - FunctionB Group - Derivative
    (A) ddx[f(g(x))]\frac{d}{dx}[f(g(x))] at x=−1x = -11. −16-16
    (B) ddx[g(f(x)−1)]\frac{d}{dx}[g(f(x) - 1)] at x=−1x = -12. 2020
    (C) ddx[f(f(x)−3)]\frac{d}{dx}[f(f(x) - 3)] at x=2x = 23. −20-20
    (D) ddx[g(g(x))]\frac{d}{dx}[g(g(x))] at x=2x = 24. 1515
    5. 1212
  3. Suppose that the functions ff and gg and their derivatives with respect to xx have the following values at x=0x = 0 and x=1x = 1.
    xxf(x)f(x)g(x)g(x)f′(x)f'(x)g′(x)g'(x)
    0011115513\frac{1}{3}
    1133−4-4−13-\frac{1}{3}−83-\frac{8}{3}
    Misc II Q.3 (i)
    The derivative of f[g(x)]f[g(x)] w. r. t. xx at x=0x = 0 is ......
  4. Misc II Q.3 (ii)
    The derivative of g[f(x)]g[f(x)] w. r. t. xx at x=0x = 0 is ......
  5. Misc II Q.3 (iii)
    The value of [ddx[x10+f(x)]−2]x=1\left[\frac{d}{dx}[x^{10} + f(x)]^{-2}\right]_{x = 1} is ......
  6. Misc II Q.3 (iv)
    The derivative of f[(x+g(x))]f[(x + g(x))] w. r. t. xx at x=0x = 0 is ......
  7. Differentiate the following w. r. t. xx
    Misc II Q.4 (i)
    sin⁡[2tan⁡−1(1−x1+x)]\sin\left[2\tan^{-1}\left(\sqrt{\frac{1 - x}{1 + x}}\right)\right]
  8. Misc II Q.4 (ii)
    sin⁡2[cot⁡−1(1+x1−x)]\sin^2\left[\cot^{-1}\left(\sqrt{\frac{1 + x}{1 - x}}\right)\right]
  9. Misc II Q.4 (iii)
    tan⁡−1[x(3−x)1−3x]\tan^{-1}\left[\frac{\sqrt{x}(3 - x)}{1 - 3x}\right]
  10. Misc II Q.4 (iv)
    cos⁡−1(1+x−1−x2)\cos^{-1}\left(\frac{\sqrt{1 + x} - \sqrt{1 - x}}{2}\right)
  11. Misc II Q.4 (v)
    tan⁡−1(x1+6x2)+cot⁡−1(1−10x27x)\tan^{-1}\left(\frac{x}{1 + 6x^2}\right) + \cot^{-1}\left(\frac{1 - 10x^2}{7x}\right)
  12. Misc II Q.4 (vi)
    tan⁡−1[1+x2+x1+x2−x]\tan^{-1}\left[\sqrt{\frac{\sqrt{1 + x^2} + x}{\sqrt{1 + x^2} - x}}\right]
  13. Misc II Q.5 (i)
    If y+x+y−x=c\sqrt{y + x} + \sqrt{y - x} = c, then show that dydx=yx−y2x2−1\frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}.
  14. Misc II Q.5 (ii)
    If x1−y2+y1−x2=1x\sqrt{1 - y^2} + y\sqrt{1 - x^2} = 1, then show that dydx=−1−y21−x2\frac{dy}{dx} = -\sqrt{\frac{1 - y^2}{1 - x^2}}.
  15. Misc II Q.5 (iii)
    If xsin⁡(a+y)+sin⁡acos⁡(a+y)=0x\sin(a + y) + \sin a \cos(a + y) = 0, then show that dydx=sin⁡2(a+y)sin⁡a\frac{dy}{dx} = \frac{\sin^2(a + y)}{\sin a}.
  16. Misc II Q.5 (iv)
    If sin⁡y=xsin⁡(a+y)\sin y = x \sin(a + y), then show that dydx=sin⁡2(a+y)sin⁡a\frac{dy}{dx} = \frac{\sin^2(a + y)}{\sin a}.
  17. Misc II Q.5 (v)
    If x=exyx = e^{\frac{x}{y}}, then show that dydx=x−yxlog⁡x\frac{dy}{dx} = \frac{x - y}{x \log x}.
  18. Misc II Q.5 (vi)
    If y=f(x)y = f(x) is a differentiable function then show that d2xdy2=−(dydx)−3⋅d2ydx2\frac{d^2x}{dy^2} = -\left(\frac{dy}{dx}\right)^{-3} \cdot \frac{d^2y}{dx^2}.
  19. Misc II Q.6 (i)
    Differentiate tan⁡−1(1+x2−1x)\tan^{-1}\left(\frac{\sqrt{1 + x^2} - 1}{x}\right) w. r. t. tan⁡−1(2x1−x21−2x2)\tan^{-1}\left(\frac{2x\sqrt{1 - x^2}}{1 - 2x^2}\right).
  20. Misc II Q.6 (ii)
    Differentiate log⁡(1+x2+x1+x2−x)\log\left(\frac{\sqrt{1 + x^2} + x}{\sqrt{1 + x^2} - x}\right) w. r. t. cos⁡(log⁡x)\cos(\log x).
  21. Misc II Q.6 (iii)
    Differentiate tan⁡−1(1+x2−1x)\tan^{-1}\left(\frac{\sqrt{1 + x^2} - 1}{x}\right) w. r. t. cos⁡−1(1+1+x221+x2)\cos^{-1}\left(\sqrt{\frac{1 + \sqrt{1 + x^2}}{2\sqrt{1 + x^2}}}\right).
  22. Misc II Q.7 (i)
    If y2=a2cos⁡2x+b2sin⁡2xy^2 = a^2\cos^2 x + b^2\sin^2 x, show that y+d2ydx2=a2b2y3y + \frac{d^2y}{dx^2} = \frac{a^2 b^2}{y^3}.
  23. Misc II Q.7 (ii)
    If log⁡y=log⁡(sin⁡x)−x2\log y = \log(\sin x) - x^2, show that d2ydx2+4xdydx+(4x2+3)y=0\frac{d^2y}{dx^2} + 4x\frac{dy}{dx} + (4x^2 + 3)y = 0.
  24. Misc II Q.7 (iii)
    If x=acos⁡θx = a\cos\theta, y=bsin⁡θy = b\sin\theta, show that a2[yd2ydx2+(dydx)2]+b2=0a^2\left[y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right] + b^2 = 0.
  25. Misc II Q.7 (iv)
    If y=Acos⁡(log⁡x)+Bsin⁡(log⁡x)y = A\cos(\log x) + B\sin(\log x), show that x2y2+xy1+y=0x^2 y_2 + x y_1 + y = 0.
  26. Misc II Q.7 (v)
    If y=Aemx+Benxy = Ae^{mx} + Be^{nx}, show that y2−(m+n)y1+(mn)y=0y_2 - (m + n)y_1 + (mn)y = 0.