Mathematics · Textbook solutions

Differentiation

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 87 questions

9.1 Derivatives from First Principle

6 q

Worked Examples

Worked · 6
  1. 9.1.4a SolvedEx.1
    Find the derivative of xnx^{n} w. r. t. xx for n∈Nn \in N.
  2. 9.1.4a SolvedEx.2
    Find derivative of sin⁡x\sin x w. r. t. xx.
  3. 9.1.4a SolvedEx.3
    Find the derivative of tan⁡x\tan x w. r. t. xx.
  4. 9.1.4a SolvedEx.4
    Find the derivative of sec⁡x\sec x w. r. t. xx.
  5. 9.1.4a SolvedEx.5
    Find the derivative of log⁡x\log x w. r. t. xx. (x>0)(x > 0)
  6. 9.1.4a SolvedEx.6
    Find then derivative of axa^{x} w. r. t. xx. (a>0)(a > 0)

9.1 Derivatives of Standard Functions

6 q

Solved Examples

Worked · 6
  1. Find the derivatives of the following from the definition,
    9.1.4 SolvedEx.1(i)
    x\sqrt{x}
  2. 9.1.4 SolvedEx.1(ii)
    cos⁡(2x+3)\cos(2x+3)
  3. 9.1.4 SolvedEx.1(iii)
    4x4^{x}
  4. 9.1.4 SolvedEx.1(iv)
    log⁡(3x−2)\log(3x-2)
  5. 9.1.4 SolvedEx.2
    Find the derivative of f(x)=sin⁡xf(x) = \sin x, at x=πx = \pi
  6. 9.1.4 SolvedEx.3
    Find the derivative of x2+x+2x^{2}+x+2, at x=−3x = -3

9.1.5 Differentiability and Continuity

4 q

Solved Examples

Worked · 4
  1. 9.1.5 SolvedEx.1
    Test whether the function f(x)=(3x−2)25f(x)=(3x-2)^{\frac{2}{5}} is differentiable at x=23x=\frac{2}{3}.
  2. 9.1.5 SolvedEx.2
    Examine the differentiability of f(x)=(x−2)∣x−2∣f(x)=(x-2)|x-2| at x=2x=2.
  3. 9.1.5 SolvedEx.3
    Show the function f(x)f(x) is continuous at x=3x=3, but not differentiable at x=3x=3. if f(x)=2x+1f(x)=2x+1 for x≤3x\le 3     =16−x2\quad\quad\;\; =16-x^2 for x>3x>3.
  4. 9.1.5 SolvedEx.4
    Show that the function f(x)f(x) is differentiable at x=−3x=-3 where, f(x)=x2+2f(x)=x^2+2.

Exercise 9.1

22 q
  1. Find the derivatives of the following w. r. t. xx by using method of first principle.
    Ex 9.1 Q1(a)
    x2+3x−1x^2+3x-1
  2. Ex 9.1 Q1(b)
    sin⁡(3x)\sin(3x)
  3. Ex 9.1 Q1(c)
    e2x+1e^{2x+1}
  4. Ex 9.1 Q1(d)
    3x3^{x}
  5. Ex 9.1 Q1(e)
    log⁡(2x+5)\log(2x+5)
  6. Ex 9.1 Q1(f)
    tan⁡(2x+3)\tan(2x+3)
  7. Ex 9.1 Q1(g)
    sec⁡(5x−2)\sec(5x-2)
  8. Ex 9.1 Q1(h)
    xxx\sqrt{x}
  9. Find the derivatives of the following w. r. t. xx. at the points indicated against them by using method of first principle
    Ex 9.1 Q2(a)
    2x+5\sqrt{2x+5} at x=2x=2
  10. Ex 9.1 Q2(b)
    tan⁡x\tan x at x=π/4x=\pi/4
  11. Ex 9.1 Q2(c)
    23x+12^{3x+1} at x=2x=2
  12. Ex 9.1 Q2(d)
    log⁡(2x+1)\log(2x+1) at x=2x=2
  13. Ex 9.1 Q2(e)
    e3x−4e^{3x-4} at x=2x=2
  14. Ex 9.1 Q2(f)
    cos⁡x\cos x at x=5π4x=\frac{5\pi}{4}
  15. Ex 9.1 Q3
    Show that the function ff is not differentiable at x=−3x=-3, where f(x)=x2+2f(x)=x^2+2 for x<−3x<-3     =2−3x\quad\quad\quad\;\; =2-3x for x≥−3x\ge -3
  16. Ex 9.1 Q4
    Show that f(x)=x2f(x)=x^2 is continuous and differentiable at x=0x=0.
  17. Discuss the continuity and differentiability of
    Ex 9.1 Q5(i)
    f(x)=x∣x∣f(x)=x|x| at x=0x=0
  18. Ex 9.1 Q5(ii)
    f(x)=(2x+3)∣2x+3∣f(x)=(2x+3)|2x+3| at x=−3/2x=-3/2
  19. Ex 9.1 Q6
    Discuss the continuity and differentiability of f(x)f(x) at x=2x=2 f(x)=[x]f(x)=[x] if x∈[0,4)x\in[0,4). [where [∗][*] is a greatest integer ( floor ) function]
  20. Ex 9.1 Q7
    Test the continuity and differentiability of f(x)=3x+2f(x)=3x+2 if x>2x>2     =12−x2\quad\quad\;\; =12-x^2 if x≤2x\le 2 at x=2x=2.
  21. Ex 9.1 Q8
    If f(x)=sin⁡x−cos⁡xf(x)=\sin x-\cos x if x≤π/2x\le \pi/2     =2x−π+1\quad\quad\;\; =2x-\pi+1 if x>π/2x>\pi/2. Test the continuity and differentiability of ff at x=π/2x=\pi/2
  22. Ex 9.1 Q9
    Examine the function f(x)=x2cos⁡(1x)f(x)=x^2\cos\left(\frac{1}{x}\right), for x≠0x\ne 0     =0\quad\quad\;\; =0, for x=0x=0 for continuity and differentiability at x=0x=0.

9.2 Rules of Differentiation

5 q

Solved Examples

Worked · 5
  1. Find the derivatives of the following functions
    9.2.4 SolvedEx.1(1)
    y=x32+log⁡x−cos⁡xy=x^{\frac{3}{2}}+\log x-\cos x
  2. 9.2.4 SolvedEx.1(2)
    f(x)=x5cosec⁡x+xtan⁡xf(x)=x^5\operatorname{cosec}x+\sqrt{x}\tan x
  3. 9.2.4 SolvedEx.1(3)
    y=ex−5ex+5y=\frac{e^x-5}{e^x+5}
  4. 9.2.4 SolvedEx.1(4)
    y=xsin⁡xx+sin⁡xy=\frac{x\sin x}{x+\sin x}
  5. 9.2.4 SolvedEx.2
    If f(x)=ptan⁡x+qsin⁡x+rf(x)=p\tan x+q\sin x+r, f(0)=−43f(0)=-4\sqrt{3}, f(π3)=−73f\left(\frac{\pi}{3}\right)=-7\sqrt{3}, f′(π3)=3f'\left(\frac{\pi}{3}\right)=3 then find pp, qq and rr.

Exercise 9.2

26 q

Exercise 9.2 (I)

Practice · 6
  1. Ex 9.2 I Q1
    Differentiate the following w.r.t. xx : y=x43+ex−sin⁡xy=x^{\frac{4}{3}}+e^{x}-\sin x
  2. Ex 9.2 I Q2
    Differentiate the following w.r.t. xx : y=x+tan⁡x−x3y=\sqrt{x}+\tan x-x^{3}
  3. Ex 9.2 I Q3
    Differentiate the following w.r.t. xx : y=log⁡x−cosec⁡x+5x−3x32y=\log x-\operatorname{cosec} x+5^{x}-\frac{3}{x^{\frac{3}{2}}}
  4. Ex 9.2 I Q4
    Differentiate the following w.r.t. xx : y=x73+5x45−5x25y=x^{\frac{7}{3}}+5x^{\frac{4}{5}}-\frac{5}{x^{\frac{2}{5}}}
  5. Ex 9.2 I Q5
    Differentiate the following w.r.t. xx : y=7x+x7−23xx−log⁡x+77y=7^{x}+x^{7}-\frac{2}{3}x\sqrt{x}-\log x+7^{7}
  6. Ex 9.2 I Q6
    Differentiate the following w.r.t. xx : y=3cot⁡x−5ex+3log⁡x−4x34y=3\cot x-5e^{x}+3\log x-\frac{4}{x^{\frac{3}{4}}}

Exercise 9.2 (II)

Practice · 6
  1. Ex 9.2 II Q1
    Differentiate the following w.r.t. xx : y=x5tan⁡xy=x^{5}\tan x
  2. Ex 9.2 II Q2
    Differentiate the following w.r.t. xx : y=x3log⁡xy=x^{3}\log x
  3. Ex 9.2 II Q3
    Differentiate the following w.r.t. xx : y=(x2+2)2sin⁡xy=(x^{2}+2)^{2}\sin x
  4. Ex 9.2 II Q4
    Differentiate the following w.r.t. xx : y=exlog⁡xy=e^{x}\log x
  5. Ex 9.2 II Q5
    Differentiate the following w.r.t. xx : y=x32exlog⁡xy=x^{\frac{3}{2}}e^{x}\log x
  6. Ex 9.2 II Q6
    Differentiate the following w.r.t. xx : y=log⁡ex3log⁡x3y=\log e^{x^{3}}\log x^{3}

Exercise 9.2 (III)

Practice · 6
  1. Ex 9.2 III Q1
    Differentiate the following w.r.t. xx : y=x2x+x4log⁡xy=x^{2}\sqrt{x}+x^{4}\log x
  2. Ex 9.2 III Q2
    Differentiate the following w.r.t. xx : y=exsec⁡x−x53log⁡xy=e^{x}\sec x-x^{\frac{5}{3}}\log x
  3. Ex 9.2 III Q3
    Differentiate the following w.r.t. xx : y=x4+xxcos⁡x−x2exy=x^{4}+x\sqrt{x}\cos x-x^{2}e^{x}
  4. Ex 9.2 III Q4
    Differentiate the following w.r.t. xx : y=(x3−2)tan⁡x−xcos⁡x+7x⋅x7y=(x^{3}-2)\tan x-x\cos x+7^{x}\cdot x^{7}
  5. Ex 9.2 III Q5
    Differentiate the following w.r.t. xx : y=sin⁡xlog⁡x+excos⁡x−exxy=\sin x\log x+e^{x}\cos x-e^{x}\sqrt{x}
  6. Ex 9.2 III Q6
    Differentiate the following w.r.t. xx : y=extan⁡x+cos⁡xlog⁡x−x 5xy=e^{x}\tan x+\cos x\log x-\sqrt{x}\,5^{x}

Exercise 9.2 (IV)

Practice · 6
  1. Ex 9.2 IV Q1
    Differentiate the following w.r.t. xx : y=x2+3x2−5y=\frac{x^{2}+3}{x^{2}-5}
  2. Ex 9.2 IV Q2
    Differentiate the following w.r.t. xx : y=x+5x−5y=\frac{\sqrt{x}+5}{\sqrt{x}-5}
  3. Ex 9.2 IV Q3
    Differentiate the following w.r.t. xx : y=xexx+exy=\frac{xe^{x}}{x+e^{x}}
  4. Ex 9.2 IV Q4
    Differentiate the following w.r.t. xx : y=xlog⁡xx+log⁡xy=\frac{x\log x}{x+\log x}
  5. Ex 9.2 IV Q5
    Differentiate the following w.r.t. xx : y=x2sin⁡xx+cos⁡xy=\frac{x^{2}\sin x}{x+\cos x}
  6. Ex 9.2 IV Q6
    Differentiate the following w.r.t. xx : y=5ex−43ex−2y=\frac{5e^{x}-4}{3e^{x}-2}

Exercise 9.2 (V)

Practice · 2
  1. Ex 9.2 V Q1
    If f(x)f(x) is a quadratic polynomial such that f(0)=3f(0)=3, f′(2)=2f'(2)=2 and f′(3)=12f'(3)=12 then find f(x)f(x).
  2. Ex 9.2 V Q2
    If f(x)=asin⁡x−bcos⁡xf(x)=a\sin x-b\cos x, f′(π4)=2f'\left(\frac{\pi}{4}\right)=\sqrt{2} and f′(π6)=2f'\left(\frac{\pi}{6}\right)=2, then find f(x)f(x).

Miscellaneous Exercise 9

18 q

(I) Select the appropriate option

Practice · 8
  1. Misc I Q1
    If y=x−4x+2y=\frac{x-4}{\sqrt{x+2}}, then dydx=\frac{dy}{dx}=
    1. A.
      1x+4\frac{1}{x+4}
    2. B.
      x(x+2)2\frac{\sqrt{x}}{\left(\sqrt{x+2}\right)^{2}}
    3. C.
      12x\frac{1}{2\sqrt{x}}
    4. D.
      x(x+2)2\frac{x}{\left(\sqrt{x}+2\right)^{2}}
  2. Misc I Q2
    If y=ax+bcx+dy=\frac{ax+b}{cx+d}, then dydx=\frac{dy}{dx}=
    1. A.
      ab−cd(cx+d)2\frac{ab-cd}{(cx+d)^{2}}
    2. B.
      ax−c(cx+d)2\frac{ax-c}{(cx+d)^{2}}
    3. C.
      ac−bd(cx+d)2\frac{ac-bd}{(cx+d)^{2}}
    4. D.
      ad−bc(cx+d)2\frac{ad-bc}{(cx+d)^{2}}
  3. Misc I Q3
    If y=3x+54x+5y=\frac{3x+5}{4x+5}, then dydx=\frac{dy}{dx}=
    1. A.
      −15(3x+5)2-\frac{15}{(3x+5)^{2}}
    2. B.
      −15(4x+5)2-\frac{15}{(4x+5)^{2}}
    3. C.
      −5(4x+5)2-\frac{5}{(4x+5)^{2}}
    4. D.
      −13(4x+5)2-\frac{13}{(4x+5)^{2}}
  4. Misc I Q4
    If y=5sin⁡x−24sin⁡x+3y=\frac{5\sin x-2}{4\sin x+3}, then dydx=\frac{dy}{dx}=
    1. A.
      7cos⁡x(4sin⁡x+3)2\frac{7\cos x}{(4\sin x+3)^{2}}
    2. B.
      23cos⁡x(4sin⁡x+3)2\frac{23\cos x}{(4\sin x+3)^{2}}
    3. C.
      −7cos⁡x(4sin⁡x+3)2-\frac{7\cos x}{(4\sin x+3)^{2}}
    4. D.
      −15cos⁡x(4sin⁡x+3)2-\frac{15\cos x}{(4\sin x+3)^{2}}
  5. Misc I Q5
    Suppose f(x)f(x) is the derivative of g(x)g(x) and g(x)g(x) is the derivative of h(x)h(x). If h(x)=asin⁡x+bcos⁡x+ch(x)=a\sin x+b\cos x+c then f(x)+h(x)=f(x)+h(x)=
    1. A.
      00
    2. B.
      cc
    3. C.
      −c-c
    4. D.
      −2(asin⁡+bcos⁡x)-2(a\sin +b\cos x)
  6. Misc I Q6
    If f(x)=2x+6f(x)=2x+6 for 0≤x≤20\le x\le 2 =ax2+bx=ax^{2}+bx for 2<x≤42<x\le 4 is differentiable at x=2x=2 then the values of aa and bb are.
    1. A.
      a=−32, b=3a=-\frac{3}{2},\ b=3
    2. B.
      a=32, b=8a=\frac{3}{2},\ b=8
    3. C.
      a=12, b=8a=\frac{1}{2},\ b=8
    4. D.
      a=−32, b=8a=-\frac{3}{2},\ b=8
  7. Misc I Q7
    If f(x)=x2+sin⁡x+1f(x)=x^{2}+\sin x+1 for x≤0x\le 0 =x2−2x+1=x^{2}-2x+1 for x≤0x\le 0 then
    1. A.
      f is continuous at x=0x=0, but not differentiable at x=0x=0
    2. B.
      f is neither continuous nor differentiable at x=0x=0
    3. C.
      f is not continuous at x=0x=0, but differentiable at x=0x=0
    4. D.
      f is both continuous and differentiable at x=0x=0
  8. Misc I Q8
    If, f(x)=x5050+x4949+x4848+....+x22+x+1f(x)=\frac{x^{50}}{50}+\frac{x^{49}}{49}+\frac{x^{48}}{48}+....+\frac{x^{2}}{2}+x+1, then f′(1)=f'(1)=
    1. A.
      4848
    2. B.
      4949
    3. C.
      5050
    4. D.
      5151

(II) Answer the following

Practice · 10
  1. Misc II Q1
    Determine whether the following function is differentiable at x=3x=3 where, f(x)=x2+2f(x)=x^{2}+2, for x≥3x\ge 3 =6x−7=6x-7, for x<3x<3.
  2. Misc II Q2
    Find the values of pp and qq that make function f(x)f(x) differentiable everywhere on RR f(x)=3−xf(x)=3-x, for x<1x<1 =px2+qx=px^{2}+qx, for x≥1x\ge 1.
  3. Misc II Q3
    Determine the values of pp and qq that make the function f(x)f(x) differentiable on RR where f(x)=px3f(x)=px^{3}, for x<2x<2 =x2+q=x^{2}+q, for x≥2x\ge 2.
  4. Misc II Q4
    Determine all real values of pp and qq that ensure the function f(x)=px+qf(x)=px+q for, x≤1x\le 1 =tan⁡(πx4)=\tan\left(\frac{\pi x}{4}\right), for 1<x<21<x<2 is differentiable at x=1x=1.
  5. Misc II Q5
    Discuss whether the function f(x)=∣x+1∣+∣x−1∣f(x)=\left|x+1\right|+\left|x-1\right| is differentiable ∀x∈R\forall x\in R
  6. Misc II Q6
    Test whether the function f(x)=2x−3f(x)=2x-3, for x≥2x\ge 2 =x−1=x-1, for x<2x<2 is differentiable at x=2x=2.
  7. Misc II Q7
    Test whether the function f(x)=x2+1f(x)=x^{2}+1, for x≥2x\ge 2 =2x+1=2x+1, for x<2x<2 is differentiable at x=2x=2.
  8. Misc II Q8
    Test whether the function f(x)=5x−3x2f(x)=5x-3x^{2} for x≥1x\ge 1 =3−x=3-x, for x<1x<1 is differentiable at x=1x=1.
  9. Misc II Q9
    If f(2)=4f(2)=4, f′(2)=1f'(2)=1 then find lim⁡x→2[xf(2)−2f(x)x−2]\lim\limits_{x\to 2}\left[\frac{xf(2)-2f(x)}{x-2}\right]
  10. Misc II Q10
    If y=exxy=\frac{e^{x}}{\sqrt{x}} find dydx\frac{dy}{dx} when x=1x=1.