Differentiation Questions


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Differentiation Questions

Differentiation questions with answers are provided here for students of Class 11 and Class 12. Differentiation is an important topic for 11th and 12th standard students as these concepts are further included in higher studies. The problems prepared here are as per the CBSE board and NCERT curriculum. Practising these questions will help students to solve hard problems and to score more marks in the exam.

The differentiation of a function f(x) is represented as f’(x). If f(x) = y, then f’(x) = dy/dx, which means y is differentiated with respect to x. Before we start solving some questions based on differentiation, let us see the general differentiation formulas used here.

Function f(x) = y Differentiation of function f’(x) = dy/dx
xn nxn-1
ex ex
ln(x) 1/x
Sin x Cos x
Cos x -sin x
Tan x sec2x
K (constant) 0

Questions on Differentiation (With Answers)

Here are a few solved questions based on differentiation concept.

1. Differentiate x5 with respect to x.

Solution: Given, y = x5

On differentiating w.r.t we get;

dy/dx = d(x5)/dx

y’ = 5x5-1 = 5x4

Therefore, d(x5)/dx = 5x4

2. Differentiate 10x2 with respect to x.

Solution: y = 10x2

y’ = d(10x2)/dx

y’ = 2.10.x = 20x

Therefore, d(10x2)/dx = 20 x

3. Differentiate 20x-4 + 9.

Solution: y = 20x-4 + 9

y’ = d(20x-4 + 9)/dx

y’ = d(20x-4)/dx + d(9)/dx

y’ = -4.20.x-4-1+0

y’ = -80x-5

Therefore, d(20x-4 + 9)/dx = -80x-5

4. Differentiate ln(10).


Since ln(10) is a constant, the derivative of ln(10) with respect to the variable “x” is 0.

Rules of Differentiation:

  • Sum and Difference : (u(x) ± v(x))’=u'(x)±v'(x)
  • Product rule: (u(x) × v(x))’=u′(x)×v(x)+u(x)×v′(x)
  • Quotient Rule : \(\frac{u(x)}{v(x)} = \frac{u^{\prime}(x) \times v(x)-u(x) \times v^{\prime}(x)}{v(x)^{2}}\)
  • Chain Rule: dy(u(x))/dx = dy/du × du/dx

Also, read:

5. Differentiate sin(3x+5)

Solution: Say, y = sin (3x+5)

dy/dx = d[sin(3x+5)]/dx

= cos (3x+5) d(3x+5)/dx [ By chain rule]

= cos (3x+5) [3]

y’ = 3 cos (3x+5)

d[sin(3x+5)]/dx = 3 cos (3x+5)

6. Differentiate tan2x.

Solution: Say, y = tan2x

dy/dx = d(tan2x)/dx

= 2tan2-1x. d(tan x)/dx

= 2tan x sec2x

d[tan2x]/dx = 2tan x sec2x

7. Compute the derivative of f(x) = sin2x.

Solution: f(x) = sin2x = sin x sin x

= d(sin x)/dx. sin x + sin x.d(sin x)/dx

= cos x. sin x + sin x cos x

= 2 sin x cos x

Practice Questions

Here are some more questions which students can practice.

Differentiate the following:

1. etan x                                        [Ans: sec2x]

2. Sin2 (2x + 1)                   [Ans: 2sin(4x+2)]

3. log7 (2x – 3)                   [Ans: 2/{(2x-3)log 7}]

4. logx 3                               [Ans: -1/{xlog3(log3x)2}]

5. 3xlogx                                     [Ans: 3xlogx.log 3(1+log x)]

6. (x+1)/x                           [Ans: 1 – (1/x)2]

7. (x-a)(x-b)                       [Ans: 2x-(b+a)]

8. 3cot x+5cosec x            [Ans: −cosec x(3cosec x+5cot x)

9. (x+cos x)/tan x             [Ans: \(\frac{\tan x-\tan x \sin x-x \sec ^{2} x-\sec x}{\tan ^{2} x}\)]

10. 1/(ax2+bx+c)              [Ans: \(\frac{-(2 a x+b)}{\left(a x^{2}+b c+c\right)^{2}}\)]