Mathematics · Textbook solutions

Continuity and Differentiability

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 191 questions

5.2 Continuity

59 q

Solved Examples

Worked · 20
  1. 5.1 Eg.1
    Check the continuity of the function ff given by f(x)=2x+3f(x) = 2x + 3 at x=1x = 1.
  2. 5.1 Eg.2
    Examine whether the function ff given by f(x)=x2f(x) = x^2 is continuous at x=0x = 0.
  3. 5.1 Eg.3
    Discuss the continuity of the function ff given by f(x)=∣x∣f(x) = |x| at x=0x = 0.
  4. 5.1 Eg.4
    Show that the function ff given by f(x)={x3+3,if x≠01,if x=0f(x) = \begin{cases} x^3 + 3, & \text{if } x \ne 0 \\ 1, & \text{if } x = 0 \end{cases} is not continuous at x=0x = 0.
  5. 5.1 Eg.5
    Check the points where the constant function f(x)=kf(x) = k is continuous.
  6. 5.1 Eg.6
    Prove that the identity function on real numbers given by f(x)=xf(x) = x is continuous at every real number.
  7. 5.1 Eg.7
    Is the function defined by f(x)=∣x∣f(x) = |x|, a continuous function?
  8. 5.1 Eg.8
    Discuss the continuity of the function ff given by f(x)=x3+x2−1f(x) = x^3 + x^2 - 1.
  9. 5.1 Eg.9
    Discuss the continuity of the function ff defined by f(x)=1xf(x) = \dfrac{1}{x}, x≠0x \ne 0.
  10. 5.1 Eg.10
    Discuss the continuity of the function ff defined by f(x)={x+2,if x≤1x−2,if x>1f(x) = \begin{cases} x + 2, & \text{if } x \le 1 \\ x - 2, & \text{if } x > 1 \end{cases}
  11. 5.1 Eg.11
    Find all the points of discontinuity of the function ff defined by f(x)={x+2,if x<10,if x=1x−2,if x>1f(x) = \begin{cases} x + 2, & \text{if } x < 1 \\ 0, & \text{if } x = 1 \\ x - 2, & \text{if } x > 1 \end{cases}
  12. 5.1 Eg.12
    Discuss the continuity of the function defined by f(x)={x+2,if x<0−x+2,if x>0f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ -x + 2, & \text{if } x > 0 \end{cases}
  13. 5.1 Eg.13
    Discuss the continuity of the function ff given by f(x)={x,if x≥0x2,if x<0f(x) = \begin{cases} x, & \text{if } x \ge 0 \\ x^2, & \text{if } x < 0 \end{cases}
  14. 5.1 Eg.14
    Show that every polynomial function is continuous.
  15. 5.1 Eg.15
    Find all the points of discontinuity of the greatest integer function defined by f(x)=[x]f(x) = [x], where [x][x] denotes the greatest integer less than or equal to xx.
  16. 5.1 Eg.16
    Prove that every rational function is continuous.
  17. 5.1 Eg.17
    Discuss the continuity of sine function.
  18. 5.1 Eg.18
    Prove that the function defined by f(x)=tan⁡xf(x) = \tan x is a continuous function.
  19. 5.1 Eg.19
    Show that the function defined by f(x)=sin⁡(x2)f(x) = \sin(x^2) is a continuous function.
  20. 5.1 Eg.20
    Show that the function ff defined by f(x)=∣1−x+∣x∣∣f(x) = |1 - x + |x||, where xx is any real number, is a continuous function.

Exercise 5.1

Practice · 39
  1. Ex 5.1 Q1
    Prove that the function f(x)=5x−3f(x) = 5x - 3 is continuous at x=0x = 0, at x=−3x = -3 and at x=5x = 5.
  2. Ex 5.1 Q2
    Examine the continuity of the function f(x)=2x2−1f(x) = 2x^2 - 1 at x=3x = 3.
  3. Examine the following functions for continuity.
    Ex 5.1 Q3(a)
    f(x)=x−5f(x) = x - 5
  4. Ex 5.1 Q3(b)
    f(x)=1x−5f(x) = \dfrac{1}{x - 5}, x≠5x \ne 5
  5. Ex 5.1 Q3(c)
    f(x)=x2−25x+5f(x) = \dfrac{x^2 - 25}{x + 5}, x≠−5x \ne -5
  6. Ex 5.1 Q3(d)
    f(x)=∣x−5∣f(x) = |x - 5|
  7. Ex 5.1 Q4
    Prove that the function f(x)=xnf(x) = x^n is continuous at x=nx = n, where nn is a positive integer.
  8. Ex 5.1 Q5
    Is the function ff defined by f(x)={x,if x≤15,if x>1f(x) = \begin{cases} x, & \text{if } x \le 1 \\ 5, & \text{if } x > 1 \end{cases} continuous at x=0x = 0? At x=1x = 1? At x=2x = 2?
  9. Ex 5.1 Q6
    Find all points of discontinuity of ff, where ff is defined by f(x)={2x+3,if x≤22x−3,if x>2f(x) = \begin{cases} 2x + 3, & \text{if } x \le 2 \\ 2x - 3, & \text{if } x > 2 \end{cases}
  10. Ex 5.1 Q7
    Find all points of discontinuity of ff, where ff is defined by f(x)={∣x∣+3,if x≤−3−2x,if −3<x<36x+2,if x≥3f(x) = \begin{cases} |x| + 3, & \text{if } x \le -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x + 2, & \text{if } x \ge 3 \end{cases}
  11. Ex 5.1 Q8
    Find all points of discontinuity of ff, where ff is defined by f(x)={∣x∣x,if x≠00,if x=0f(x) = \begin{cases} \dfrac{|x|}{x}, & \text{if } x \ne 0 \\ 0, & \text{if } x = 0 \end{cases}
  12. Ex 5.1 Q9
    Find all points of discontinuity of ff, where ff is defined by f(x)={x∣x∣,if x<0−1,if x≥0f(x) = \begin{cases} \dfrac{x}{|x|}, & \text{if } x < 0 \\ -1, & \text{if } x \ge 0 \end{cases}
  13. Ex 5.1 Q10
    Find all points of discontinuity of ff, where ff is defined by f(x)={x+1,if x≥1x2+1,if x<1f(x) = \begin{cases} x + 1, & \text{if } x \ge 1 \\ x^2 + 1, & \text{if } x < 1 \end{cases}
  14. Ex 5.1 Q11
    Find all points of discontinuity of ff, where ff is defined by f(x)={x3−3,if x≤2x2+1,if x>2f(x) = \begin{cases} x^3 - 3, & \text{if } x \le 2 \\ x^2 + 1, & \text{if } x > 2 \end{cases}
  15. Ex 5.1 Q12
    Find all points of discontinuity of ff, where ff is defined by f(x)={x10−1,if x≤1x2,if x>1f(x) = \begin{cases} x^{10} - 1, & \text{if } x \le 1 \\ x^2, & \text{if } x > 1 \end{cases}
  16. Ex 5.1 Q13
    Is the function defined by f(x)={x+5,if x≤1x−5,if x>1f(x) = \begin{cases} x + 5, & \text{if } x \le 1 \\ x - 5, & \text{if } x > 1 \end{cases} a continuous function?
  17. Ex 5.1 Q14
    Discuss the continuity of the function ff, where ff is defined by f(x)={3,if 0≤x≤14,if 1<x<35,if 3≤x≤10f(x) = \begin{cases} 3, & \text{if } 0 \le x \le 1 \\ 4, & \text{if } 1 < x < 3 \\ 5, & \text{if } 3 \le x \le 10 \end{cases}
  18. Ex 5.1 Q15
    Discuss the continuity of the function ff, where ff is defined by f(x)={2x,if x<00,if 0≤x≤14x,if x>1f(x) = \begin{cases} 2x, & \text{if } x < 0 \\ 0, & \text{if } 0 \le x \le 1 \\ 4x, & \text{if } x > 1 \end{cases}
  19. Ex 5.1 Q16
    Discuss the continuity of the function ff, where ff is defined by f(x)={−2,if x≤−12x,if −1<x≤12,if x>1f(x) = \begin{cases} -2, & \text{if } x \le -1 \\ 2x, & \text{if } -1 < x \le 1 \\ 2, & \text{if } x > 1 \end{cases}
  20. Ex 5.1 Q17
    Find the relationship between aa and bb so that the function ff defined by f(x)={ax+1,if x≤3bx+3,if x>3f(x) = \begin{cases} ax + 1, & \text{if } x \le 3 \\ bx + 3, & \text{if } x > 3 \end{cases} is continuous at x=3x = 3.
  21. Ex 5.1 Q18
    For what value of λ\lambda is the function defined by f(x)={λ(x2−2x),if x≤04x+1,if x>0f(x) = \begin{cases} \lambda(x^2 - 2x), & \text{if } x \le 0 \\ 4x + 1, & \text{if } x > 0 \end{cases} continuous at x=0x = 0? What about continuity at x=1x = 1?
  22. Ex 5.1 Q19
    Show that the function defined by g(x)=x−[x]g(x) = x - [x] is discontinuous at all integral points. Here [x][x] denotes the greatest integer less than or equal to xx.
  23. Ex 5.1 Q20
    Is the function defined by f(x)=x2−sin⁡x+5f(x) = x^2 - \sin x + 5 continuous at x=πx = \pi?
  24. Discuss the continuity of the following functions:
    Ex 5.1 Q21(a)
    f(x)=sin⁡x+cos⁡xf(x) = \sin x + \cos x
  25. Ex 5.1 Q21(b)
    f(x)=sin⁡x−cos⁡xf(x) = \sin x - \cos x
  26. Ex 5.1 Q21(c)
    f(x)=sin⁡x⋅cos⁡xf(x) = \sin x \cdot \cos x
  27. Ex 5.1 Q22
    Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
  28. Ex 5.1 Q23
    Find all points of discontinuity of ff, where f(x)={sin⁡xx,if x<0x+1,if x≥0f(x) = \begin{cases} \dfrac{\sin x}{x}, & \text{if } x < 0 \\ x + 1, & \text{if } x \ge 0 \end{cases}
  29. Ex 5.1 Q24
    Determine if ff defined by f(x)={x2sin⁡1x,if x≠00,if x=0f(x) = \begin{cases} x^2 \sin \dfrac{1}{x}, & \text{if } x \ne 0 \\ 0, & \text{if } x = 0 \end{cases} is a continuous function?
  30. Ex 5.1 Q25
    Examine the continuity of ff, where ff is defined by f(x)={sin⁡x−cos⁡x,if x≠0−1,if x=0f(x) = \begin{cases} \sin x - \cos x, & \text{if } x \ne 0 \\ -1, & \text{if } x = 0 \end{cases}
  31. Ex 5.1 Q26
    Find the value of kk so that the function ff is continuous at the indicated point: f(x)={kcos⁡xπ−2x,if x≠π23,if x=π2f(x) = \begin{cases} \dfrac{k \cos x}{\pi - 2x}, & \text{if } x \ne \dfrac{\pi}{2} \\ 3, & \text{if } x = \dfrac{\pi}{2} \end{cases} at x=π2x = \dfrac{\pi}{2}
  32. Ex 5.1 Q27
    Find the value of kk so that the function ff is continuous at the indicated point: f(x)={kx2,if x≤23,if x>2f(x) = \begin{cases} kx^2, & \text{if } x \le 2 \\ 3, & \text{if } x > 2 \end{cases} at x=2x = 2
  33. Ex 5.1 Q28
    Find the value of kk so that the function ff is continuous at the indicated point: f(x)={kx+1,if x≤πcos⁡x,if x>πf(x) = \begin{cases} kx + 1, & \text{if } x \le \pi \\ \cos x, & \text{if } x > \pi \end{cases} at x=πx = \pi
  34. Ex 5.1 Q29
    Find the value of kk so that the function ff is continuous at the indicated point: f(x)={kx+1,if x≤53x−5,if x>5f(x) = \begin{cases} kx + 1, & \text{if } x \le 5 \\ 3x - 5, & \text{if } x > 5 \end{cases} at x=5x = 5
  35. Ex 5.1 Q30
    Find the values of aa and bb such that the function defined by f(x)={5,if x≤2ax+b,if 2<x<1021,if x≥10f(x) = \begin{cases} 5, & \text{if } x \le 2 \\ ax + b, & \text{if } 2 < x < 10 \\ 21, & \text{if } x \ge 10 \end{cases} is a continuous function.
  36. Ex 5.1 Q31
    Show that the function defined by f(x)=cos⁡(x2)f(x) = \cos(x^2) is a continuous function.
  37. Ex 5.1 Q32
    Show that the function defined by f(x)=∣cos⁡x∣f(x) = |\cos x| is a continuous function.
  38. Ex 5.1 Q33
    Examine that sin⁡∣x∣\sin |x| is a continuous function.
  39. Ex 5.1 Q34
    Find all the points of discontinuity of ff defined by f(x)=∣x∣−∣x+1∣f(x) = |x| - |x + 1|.

5.3.1 Derivatives of Composite Functions

11 q

Solved Examples

Worked · 1
  1. 5.2 Eg.21
    Find the derivative of the function given by f(x)=sin⁡(x2)f(x) = \sin(x^2).

Exercise 5.2

Practice · 10
  1. Ex 5.2 Q1
    Differentiate the function sin⁡(x2+5)\sin(x^2 + 5) with respect to xx.
  2. Ex 5.2 Q2
    Differentiate the function cos⁡(sin⁡x)\cos(\sin x) with respect to xx.
  3. Ex 5.2 Q3
    Differentiate the function sin⁡(ax+b)\sin(ax + b) with respect to xx.
  4. Ex 5.2 Q4
    Differentiate the function sec⁡(tan⁡(x))\sec(\tan(\sqrt{x})) with respect to xx.
  5. Ex 5.2 Q5
    Differentiate the function sin⁡(ax+b)cos⁡(cx+d)\frac{\sin(ax + b)}{\cos(cx + d)} with respect to xx.
  6. Ex 5.2 Q6
    Differentiate the function cos⁡x3⋅sin⁡2(x5)\cos x^3 \cdot \sin^2(x^5) with respect to xx.
  7. Ex 5.2 Q7
    Differentiate the function 2cot⁡(x2)2\sqrt{\cot(x^2)} with respect to xx.
  8. Ex 5.2 Q8
    Differentiate the function cos⁡(x)\cos(\sqrt{x}) with respect to xx.
  9. Ex 5.2 Q9
    Prove that the function ff given by f(x)=∣x−1∣, x∈Rf(x) = |x - 1|,\ x \in \mathbf{R} is not differentiable at x=1x = 1.
  10. Ex 5.2 Q10
    Prove that the greatest integer function defined by f(x)=[x], 0<x<3f(x) = [x],\ 0 < x < 3 is not differentiable at x=1x = 1 and x=2x = 2.

5.3.2-5.3.3 Derivatives of Implicit and Inverse Trigonometric Functions

18 q

Solved Examples

Worked · 3
  1. 5.3 Eg.22
    Find dydx\frac{dy}{dx} if x−y=πx - y = \pi.
  2. 5.3 Eg.23
    Find dydx\frac{dy}{dx}, if y+sin⁡y=cos⁡xy + \sin y = \cos x.
  3. 5.3 Eg.24
    Find the derivative of ff given by f(x)=sin⁡−1xf(x) = \sin^{-1} x assuming it exists.

Exercise 5.3

Practice · 15
  1. Ex 5.3 Q1
    Find dydx\frac{dy}{dx} in the following: 2x+3y=sin⁡x2x + 3y = \sin x.
  2. Ex 5.3 Q2
    Find dydx\frac{dy}{dx} in the following: 2x+3y=sin⁡y2x + 3y = \sin y.
  3. Ex 5.3 Q3
    Find dydx\frac{dy}{dx} in the following: ax+by2=cos⁡yax + by^2 = \cos y.
  4. Ex 5.3 Q4
    Find dydx\frac{dy}{dx} in the following: xy+y2=tan⁡x+yxy + y^2 = \tan x + y.
  5. Ex 5.3 Q5
    Find dydx\frac{dy}{dx} in the following: x2+xy+y2=100x^2 + xy + y^2 = 100.
  6. Ex 5.3 Q6
    Find dydx\frac{dy}{dx} in the following: x3+x2y+xy2+y3=81x^3 + x^2y + xy^2 + y^3 = 81.
  7. Ex 5.3 Q7
    Find dydx\frac{dy}{dx} in the following: sin⁡2y+cos⁡xy=κ\sin^2 y + \cos xy = \kappa.
  8. Ex 5.3 Q8
    Find dydx\frac{dy}{dx} in the following: sin⁡2x+cos⁡2y=1\sin^2 x + \cos^2 y = 1.
  9. Ex 5.3 Q9
    Find dydx\frac{dy}{dx} in the following: y=sin⁡−1(2x1+x2)y = \sin^{-1}\left(\frac{2x}{1 + x^2}\right).
  10. Ex 5.3 Q10
    Find dydx\frac{dy}{dx} in the following: y=tan⁡−1(3x−x31−3x2)y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right), −13<x<13-\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}.
  11. Ex 5.3 Q11
    Find dydx\frac{dy}{dx} in the following: y=cos⁡−1(1−x21+x2)y = \cos^{-1}\left(\frac{1 - x^2}{1 + x^2}\right), 0<x<10 < x < 1.
  12. Ex 5.3 Q12
    Find dydx\frac{dy}{dx} in the following: y=sin⁡−1(1−x21+x2)y = \sin^{-1}\left(\frac{1 - x^2}{1 + x^2}\right), 0<x<10 < x < 1.
  13. Ex 5.3 Q13
    Find dydx\frac{dy}{dx} in the following: y=cos⁡−1(2x1+x2)y = \cos^{-1}\left(\frac{2x}{1 + x^2}\right), −1<x<1-1 < x < 1.
  14. Ex 5.3 Q14
    Find dydx\frac{dy}{dx} in the following: y=sin⁡−1(2x1−x2)y = \sin^{-1}\left(2x\sqrt{1 - x^2}\right), −12<x<12-\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}.
  15. Ex 5.3 Q15
    Find dydx\frac{dy}{dx} in the following: y=sec⁡−1(12x2−1)y = \sec^{-1}\left(\frac{1}{2x^2 - 1}\right), 0<x<120 < x < \frac{1}{\sqrt{2}}.

5.4 Exponential and Logarithmic Functions

15 q

Solved Examples

Worked · 5
  1. 5.4 Eg.25
    Is it true that x=elog⁡xx = e^{\log x} for all real xx?
  2. Differentiate the following w.r.t. xx:
    5.4 Eg.26(i)
    e−xe^{-x}
  3. 5.4 Eg.26(ii)
    sin⁡(log⁡x)\sin(\log x), x>0x > 0
  4. 5.4 Eg.26(iii)
    cos⁡−1(ex)\cos^{-1}(e^x)
  5. 5.4 Eg.26(iv)
    ecos⁡xe^{\cos x}

Exercise 5.4

Practice · 10
  1. Ex 5.4 Q1
    Differentiate exsin⁡x\dfrac{e^x}{\sin x} w.r.t. xx.
  2. Ex 5.4 Q2
    Differentiate esin⁡−1xe^{\sin^{-1} x} w.r.t. xx.
  3. Ex 5.4 Q3
    Differentiate ex3e^{x^3} w.r.t. xx.
  4. Ex 5.4 Q4
    Differentiate sin⁡(tan⁡−1e−x)\sin(\tan^{-1} e^{-x}) w.r.t. xx.
  5. Ex 5.4 Q5
    Differentiate log⁡(cos⁡ex)\log(\cos e^x) w.r.t. xx.
  6. Ex 5.4 Q6
    Differentiate ex+ex2+…+ex5e^x + e^{x^2} + \ldots + e^{x^5} w.r.t. xx.
  7. Ex 5.4 Q7
    Differentiate ex\sqrt{e^{\sqrt{x}}}, x>0x > 0, w.r.t. xx.
  8. Ex 5.4 Q8
    Differentiate log⁡(log⁡x)\log(\log x), x>1x > 1, w.r.t. xx.
  9. Ex 5.4 Q9
    Differentiate cos⁡xlog⁡x\dfrac{\cos x}{\log x}, x>0x > 0, w.r.t. xx.
  10. Ex 5.4 Q10
    Differentiate cos⁡(log⁡x+ex)\cos(\log x + e^x), x>0x > 0, w.r.t. xx.

5.5 Logarithmic Differentiation

22 q

Solved Examples

Worked · 4
  1. 5.5 Eg.27
    Differentiate (x−3)(x2+4)3x2+4x+5\sqrt{\dfrac{(x-3)(x^2+4)}{3x^2+4x+5}} w.r.t. xx.
  2. 5.5 Eg.28
    Differentiate axa^x w.r.t. xx, where aa is a positive constant.
  3. 5.5 Eg.29
    Differentiate xsin⁡xx^{\sin x}, x>0x > 0 w.r.t. xx.
  4. 5.5 Eg.30
    Find dydx\dfrac{dy}{dx}, if yx+xy+xx=aby^x + x^y + x^x = a^b.

Exercise 5.5

Practice · 18
  1. Ex 5.5 Q1
    Differentiate cos⁡x⋅cos⁡2x⋅cos⁡3x\cos x \cdot \cos 2x \cdot \cos 3x w.r.t. xx.
  2. Ex 5.5 Q2
    Differentiate (x−1)(x−2)(x−3)(x−4)(x−5)\sqrt{\dfrac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}} w.r.t. xx.
  3. Ex 5.5 Q3
    Differentiate (log⁡x)cos⁡x(\log x)^{\cos x} w.r.t. xx.
  4. Ex 5.5 Q4
    Differentiate xx−2sin⁡xx^x - 2^{\sin x} w.r.t. xx.
  5. Ex 5.5 Q5
    Differentiate (x+3)2⋅(x+4)3⋅(x+5)4(x+3)^2 \cdot (x+4)^3 \cdot (x+5)^4 w.r.t. xx.
  6. Ex 5.5 Q6
    Differentiate (x+1x)x+x(1+1x)\left(x + \dfrac{1}{x}\right)^x + x^{\left(1 + \frac{1}{x}\right)} w.r.t. xx.
  7. Ex 5.5 Q7
    Differentiate (log⁡x)x+xlog⁡x(\log x)^x + x^{\log x} w.r.t. xx.
  8. Ex 5.5 Q8
    Differentiate (sin⁡x)x+sin⁡−1x(\sin x)^x + \sin^{-1}\sqrt{x} w.r.t. xx.
  9. Ex 5.5 Q9
    Differentiate xsin⁡x+(sin⁡x)cos⁡xx^{\sin x} + (\sin x)^{\cos x} w.r.t. xx.
  10. Ex 5.5 Q10
    Differentiate xxcos⁡x+x2+1x2−1x^{x\cos x} + \dfrac{x^2+1}{x^2-1} w.r.t. xx.
  11. Ex 5.5 Q11
    Differentiate (xcos⁡x)x+(xsin⁡x)1x(x\cos x)^x + (x\sin x)^{\frac{1}{x}} w.r.t. xx.
  12. Ex 5.5 Q12
    Find dydx\dfrac{dy}{dx} of the function xy+yx=1x^y + y^x = 1.
  13. Ex 5.5 Q13
    Find dydx\dfrac{dy}{dx} of the function yx=xyy^x = x^y.
  14. Ex 5.5 Q14
    Find dydx\dfrac{dy}{dx} of the function (cos⁡x)y=(cos⁡y)x(\cos x)^y = (\cos y)^x.
  15. Ex 5.5 Q15
    Find dydx\dfrac{dy}{dx} of the function xy=e(x−y)xy = e^{(x-y)}.
  16. Ex 5.5 Q16
    Find the derivative of the function given by f(x)=(1+x)(1+x2)(1+x4)(1+x8)f(x) = (1+x)(1+x^2)(1+x^4)(1+x^8) and hence find f′(1)f'(1).
  17. Ex 5.5 Q17
    Differentiate (x2−5x+8)(x3+7x+9)(x^2 - 5x + 8)(x^3 + 7x + 9) in three ways mentioned below: (i) by using product rule (ii) by expanding the product to obtain a single polynomial. (iii) by logarithmic differentiation. Do they all give the same answer?
  18. Ex 5.5 Q18
    If uu, vv and ww are functions of xx, then show that ddx(u⋅v⋅w)=dudx v⋅w+u⋅dvdx⋅w+u⋅vdwdx\frac{d}{dx}(u \cdot v \cdot w) = \frac{du}{dx}\, v \cdot w + u \cdot \frac{dv}{dx} \cdot w + u \cdot v \frac{dw}{dx} in two ways - first by repeated application of product rule, second by logarithmic differentiation.

5.6 Derivatives of Functions in Parametric Forms

15 q

Solved Examples

Worked · 4
  1. 5.6 Eg.31
    Find dydx\frac{dy}{dx}, if x=acos⁡θx = a \cos \theta, y=asin⁡θy = a \sin \theta.
  2. 5.6 Eg.32
    Find dydx\frac{dy}{dx}, if x=at2x = at^2, y=2aty = 2at.
  3. 5.6 Eg.33
    Find dydx\frac{dy}{dx}, if x=a(θ+sin⁡θ)x = a(\theta + \sin \theta), y=a(1−cos⁡θ)y = a(1 - \cos \theta).
  4. 5.6 Eg.34
    Find dydx\frac{dy}{dx}, if x23+y23=a23x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}.

Exercise 5.6

Practice · 11
  1. Ex 5.6 Q1
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: x=2at2x = 2at^2, y=at4y = at^4.
  2. Ex 5.6 Q2
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: x=acos⁡θx = a \cos \theta, y=bcos⁡θy = b \cos \theta.
  3. Ex 5.6 Q3
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: x=sin⁡tx = \sin t, y=cos⁡2ty = \cos 2t.
  4. Ex 5.6 Q4
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: x=4tx = 4t, y=4ty = \frac{4}{t}.
  5. Ex 5.6 Q5
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: x=cos⁡θ−cos⁡2θx = \cos \theta - \cos 2\theta, y=sin⁡θ−sin⁡2θy = \sin \theta - \sin 2\theta.
  6. Ex 5.6 Q6
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: x=a(θ−sin⁡θ)x = a(\theta - \sin \theta), y=a(1+cos⁡θ)y = a(1 + \cos \theta).
  7. Ex 5.6 Q7
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: x=sin⁡3tcos⁡2tx = \frac{\sin^3 t}{\sqrt{\cos 2t}}, y=cos⁡3tcos⁡2ty = \frac{\cos^3 t}{\sqrt{\cos 2t}}.
  8. Ex 5.6 Q8
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: x=a(cos⁡t+log⁡tan⁡t2)x = a\left(\cos t + \log \tan \frac{t}{2}\right), y=asin⁡ty = a \sin t.
  9. Ex 5.6 Q9
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: x=asec⁡θx = a \sec \theta, y=btan⁡θy = b \tan \theta.
  10. Ex 5.6 Q10
    If xx and yy are connected parametrically by the equations given below, without eliminating the parameter, find dydx\frac{dy}{dx}: 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).
  11. Ex 5.6 Q11
    If x=asin⁡−1tx = \sqrt{a^{\sin^{-1} t}}, y=acos⁡−1ty = \sqrt{a^{\cos^{-1} t}}, show that dydx=−yx\frac{dy}{dx} = -\frac{y}{x}.

5.7 Second Order Derivative

21 q

Solved Examples

Worked · 4
  1. 5.7 Eg.35
    Find d2ydx2\frac{d^2y}{dx^2}, if y=x3+tan⁡xy = x^3 + \tan x.
  2. 5.7 Eg.36
    If y=Asin⁡x+Bcos⁡xy = \mathrm{A} \sin x + \mathrm{B} \cos x, then prove that d2ydx2+y=0\frac{d^2y}{dx^2} + y = 0.
  3. 5.7 Eg.37
    If y=3e2x+2e3xy = 3e^{2x} + 2e^{3x}, prove that d2ydx2−5dydx+6y=0\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 0.
  4. 5.7 Eg.38
    If y=sin⁡−1xy = \sin^{-1} x, show that (1−x2)d2ydx2−xdydx=0(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 0.

Exercise 5.7

Practice · 17
  1. Ex 5.7 Q1
    Find the second order derivative of the function x2+3x+2x^2 + 3x + 2.
  2. Ex 5.7 Q2
    Find the second order derivative of the function x20x^{20}.
  3. Ex 5.7 Q3
    Find the second order derivative of the function x⋅cos⁡xx \cdot \cos x.
  4. Ex 5.7 Q4
    Find the second order derivative of the function log⁡x\log x.
  5. Ex 5.7 Q5
    Find the second order derivative of the function x3log⁡xx^3 \log x.
  6. Ex 5.7 Q6
    Find the second order derivative of the function exsin⁡5xe^x \sin 5x.
  7. Ex 5.7 Q7
    Find the second order derivative of the function e6xcos⁡3xe^{6x} \cos 3x.
  8. Ex 5.7 Q8
    Find the second order derivative of the function tan⁡−1x\tan^{-1} x.
  9. Ex 5.7 Q9
    Find the second order derivative of the function log⁡(log⁡x)\log(\log x).
  10. Ex 5.7 Q10
    Find the second order derivative of the function sin⁡(log⁡x)\sin(\log x).
  11. Ex 5.7 Q11
    If y=5cos⁡x−3sin⁡xy = 5 \cos x - 3 \sin x, prove that d2ydx2+y=0\frac{d^2y}{dx^2} + y = 0.
  12. Ex 5.7 Q12
    If y=cos⁡−1xy = \cos^{-1} x, find d2ydx2\frac{d^2y}{dx^2} in terms of yy alone.
  13. Ex 5.7 Q13
    If y=3cos⁡(log⁡x)+4sin⁡(log⁡x)y = 3 \cos(\log x) + 4 \sin(\log x), show that x2y2+xy1+y=0x^2 y_2 + x y_1 + y = 0.
  14. Ex 5.7 Q14
    If y=Aemx+Benxy = \mathrm{A}e^{mx} + \mathrm{B}e^{nx}, show that d2ydx2−(m+n)dydx+mny=0\frac{d^2y}{dx^2} - (m + n)\frac{dy}{dx} + mny = 0.
  15. Ex 5.7 Q15
    If y=500e7x+600e−7xy = 500e^{7x} + 600e^{-7x}, show that d2ydx2=49y\frac{d^2y}{dx^2} = 49y.
  16. Ex 5.7 Q16
    If ey(x+1)=1e^y(x + 1) = 1, show that d2ydx2=(dydx)2\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2.
  17. Ex 5.7 Q17
    If y=(tan⁡−1x)2y = (\tan^{-1} x)^2, show that (x2+1)2y2+2x(x2+1)y1=2(x^2 + 1)^2 y_2 + 2x(x^2 + 1)y_1 = 2.

Miscellaneous Exercise on Chapter 5

30 q

Solved Examples

Worked · 8
  1. Differentiate w.r.t. xx, the following function:
    Misc Eg.39(i)
    3x+2+12x2+4\sqrt{3x+2} + \dfrac{1}{\sqrt{2x^2+4}}
  2. Misc Eg.39(ii)
    log⁡7(log⁡x)\log_7(\log x)
  3. Differentiate the following w.r.t. xx.
    Misc Eg.40(i)
    cos⁡−1(sin⁡x)\cos^{-1}(\sin x)
  4. Misc Eg.40(ii)
    tan⁡−1(sin⁡x1+cos⁡x)\tan^{-1}\left(\dfrac{\sin x}{1+\cos x}\right)
  5. Misc Eg.40(iii)
    sin⁡−1(2x+11+4x)\sin^{-1}\left(\dfrac{2^{x+1}}{1+4^x}\right)
  6. Misc Eg.41
    Find f′(x)f'(x) if f(x)=(sin⁡x)sin⁡xf(x) = (\sin x)^{\sin x} for all 0<x<π0 < x < \pi.
  7. Misc Eg.42
    For a positive constant aa find dydx\dfrac{dy}{dx}, where y=at+1ty = a^{t+\frac{1}{t}}, and x=(t+1t)ax = \left(t+\dfrac{1}{t}\right)^a.
  8. Misc Eg.43
    Differentiate sin⁡2x\sin^2 x w.r.t. ecos⁡xe^{\cos x}.

Miscellaneous Exercise

Practice · 22
  1. Misc Q1
    Differentiate w.r.t. xx the function (3x2−9x+5)9(3x^2 - 9x + 5)^9.
  2. Misc Q2
    Differentiate w.r.t. xx the function sin⁡3x+cos⁡6x\sin^3 x + \cos^6 x.
  3. Misc Q3
    Differentiate w.r.t. xx the function (5x)3cos⁡2x(5x)^{3\cos 2x}.
  4. Misc Q4
    Differentiate w.r.t. xx the function sin⁡−1(xx)\sin^{-1}(x\sqrt{x}), 0≤x≤10 \le x \le 1.
  5. Misc Q5
    Differentiate w.r.t. xx the function cos⁡−1x22x+7\dfrac{\cos^{-1}\frac{x}{2}}{\sqrt{2x+7}}, −2<x<2-2 < x < 2.
  6. Misc Q6
    Differentiate w.r.t. xx the function cot⁡−1[1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x]\cot^{-1}\left[\dfrac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right], 0<x<π20 < x < \dfrac{\pi}{2}.
  7. Misc Q7
    Differentiate w.r.t. xx the function (log⁡x)log⁡x(\log x)^{\log x}, x>1x > 1.
  8. Misc Q8
    Differentiate w.r.t. xx the function cos⁡(acos⁡x+bsin⁡x)\cos(a\cos x + b\sin x), for some constant aa and bb.
  9. Misc Q9
    Differentiate w.r.t. xx the function (sin⁡x−cos⁡x)(sin⁡x−cos⁡x)(\sin x - \cos x)^{(\sin x - \cos x)}, π4<x<3π4\dfrac{\pi}{4} < x < \dfrac{3\pi}{4}.
  10. Misc Q10
    Differentiate w.r.t. xx the function xx+xa+ax+aax^x + x^a + a^x + a^a, for some fixed a>0a > 0 and x>0x > 0.
  11. Misc Q11
    Differentiate w.r.t. xx the function xx2−3+(x−3)x2x^{x^2-3} + (x-3)^{x^2}, for x>3x > 3.
  12. Misc Q12
    Find dydx\dfrac{dy}{dx}, if y=12(1−cos⁡t)y = 12(1-\cos t), x=10(t−sin⁡t)x = 10(t-\sin t), −π2<t<π2-\dfrac{\pi}{2} < t < \dfrac{\pi}{2}.
  13. Misc Q13
    Find dydx\dfrac{dy}{dx}, if y=sin⁡−1x+sin⁡−11−x2y = \sin^{-1} x + \sin^{-1}\sqrt{1-x^2}, 0<x<10 < x < 1.
  14. Misc Q14
    If x1+y+y1+x=0x\sqrt{1+y} + y\sqrt{1+x} = 0, for , −1<x<1-1 < x < 1, prove that dydx=−1(1+x)2\dfrac{dy}{dx} = -\dfrac{1}{(1+x)^2}.
  15. Misc Q15
    If (x−a)2+(y−b)2=c2(x-a)^2 + (y-b)^2 = c^2, for some c>0c > 0, prove that [1+(dydx)2]32d2ydx2\dfrac{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} is a constant independent of aa and bb.
  16. Misc Q16
    If cos⁡y=xcos⁡(a+y)\cos y = x\cos(a+y), with cos⁡a≠±1\cos a \ne \pm 1, prove that dydx=cos⁡2(a+y)sin⁡a\dfrac{dy}{dx} = \dfrac{\cos^2(a+y)}{\sin a}.
  17. Misc Q17
    If x=a(cos⁡t+tsin⁡t)x = a(\cos t + t\sin t) and y=a(sin⁡t−tcos⁡t)y = a(\sin t - t\cos t), find d2ydx2\dfrac{d^2y}{dx^2}.
  18. Misc Q18
    If f(x)=∣x∣3f(x) = |x|^3, show that f′′(x)f''(x) exists for all real xx and find it.
  19. Misc Q19
    Using the fact that sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B) = \sin A\cos B + \cos A\sin B and the differentiation, obtain the sum formula for cosines.
  20. Misc Q20
    Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.
  21. Misc Q21
    If y=∣f(x)g(x)h(x)lmnabc∣y = \begin{vmatrix} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{vmatrix}, prove that dydx=∣f′(x)g′(x)h′(x)lmnabc∣\dfrac{dy}{dx} = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\ l & m & n \\ a & b & c \end{vmatrix}.
  22. Misc Q22
    If y=eacos⁡−1xy = e^{a\cos^{-1} x}, −1≤x≤1-1 \le x \le 1, show that (1−x2)d2ydx2−xdydx−a2y=0\left(1-x^2\right)\dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} - a^2 y = 0.