Differentiation Practice Questions with Solutions (Easy)

Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. It is used in various fields like physics, engineering, and economics to find rates of change, slopes of curves, and optimization solutions.

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Important Formula of Differentiation

Below are some basic differentiation formulas to help you solve the questions:

Check: Differentiation Formulas

Solved Practice Problem on Differentiation

The following question focuses on differentiation at an easy level.

Question 1: Differentiate f(x) = x⁵ with respect to x.

Solution:

Given, f(x)= x ⁵
dy / dx = d ( x⁵)/dx
dy / dx= 5x ^5-1= 5x ⁴

Question 2: Differentiate f(x)= 10x² with respect to x.

Solution:

Given, f(x)= 10x²
dy / dx = d ( 10x²)/dx
dy / dx= 2.10x^2-1= 20x

Question 3: Differentiate g(x) = sin x with respect to x.

Solution:

Given, g(x) = sinx
d ( sinx)/dx = cos x

Question 4: Differentiate g(x) = x2 . x3 with respect to x.

Solution:

Given , g(x) = x².x³
g(x) = x⁵
d (x⁵) / dy = 5x⁴

Question 5: Differentiate g(x) = e^x with respect to x.

Solution:

Given, g(x) = e^x
dy / dx = d ( e^x)/ dx
dy/ dx= e^x

Question 6: Differentiate f(x) = 20x^-4 + 8 with respect to x.

Solution:

Given, f(x) = 20x^-4+ 8
dy / dx = d ( 20x^-4)/ dx + d (8)/ dx
dy/dx = -4 * 20x^-4-1 +0= - 80x ^-5

Question 7: Differentiate g(x) = 3x + 5 with respect to x.

Solution:

Given, g(x) = 3x+5
dy / dx = d ( 3x)/ dx + d (5)/ dx
dy/dx = 3 + 0 = 3

Question 8: Differentiate g(t) = t³cos (t) with respect to t.

Solution:

Given ,g(t)=t³cost
We have to use product rule to find derivative

u = t³
u' = 3t²
v = cost
v'= -sint

Product Rule = uv'+vu'
g'(x) = (t³) (-sint) + (cost)(3t²)
g'(x) = -t³sint + 3t²cost
g'(x) = t²( 3 cos t -t sin t )

Question 9: Differentiate g(x) = e^xsinx with respect to x.

Solution:

Given ,g(t)=e^xsinx

We have to use product rule to find derivative

u = e^x
u' = e^x
v = sin x
v'= cos x

Product Rule = uv'+vu'
g'(x) = (e^x) (cos x) + (sinx)(e^x)
g'(x) = e^xcosx+ e^xsinx
g'(x) = e^x( cosx + sinx )

Question 10: Differentiate g(x) = sinx / 1+ cosx with respect to x.

Solution:

Given, g(x)=sinx/1+ cosx
We have to use quotient rule to find derivative

u = sin x
u' = cos x
v = 1 + cos x
v'= -sinx

Quotient Rule = u'v-uv'/v²
g'(x) = (cosx) (1+cosx) - (sinx)(-sinx) / (1 + cosx ) ²
g'(x) = cos x +cos x ² + sinx ² / (1 + cosx) ²
g'(x) = 1 +cosx / (1 + cosx ) ²
g'(x) = 1 / 1 + cosx

Question 11: Differentiate f(x) =(x²+ 1)³ with respect to x.

Solution:

Given, f(x)=(x²+1)³
We have to use chain rule to find derivative

Chain Rule = f'(g(x)).g'(x)
f'(x) = 3.(x ²+ 1)^3-1. 2x
f'(x) = 3 (x ²+1) ².2x
f'(x) = 6x (x² + 1 ) ²

Question 12: Differentiate f(x) =(3x+2)⁴ with respect to x.

Solution:

Given , f(x) = (3x+2) ⁴
We have to use chain rule to find derivative

Chain Rule = f'(g(x)).g'(x)
f'(x) = 4.(3x+2)^4-1.3
f'(x) = 4.(3x+2)³.3
f'(x) = 12 ( 3x + 2 )³

Unsolved Practice Question on Differentiation

Question 1: Differentiate f(x) = x⁷ with respect to x.

Question 2: Differentiate f(x) = 5x with respect to x.

Question 3: Differentiate f(x) = cos x with respect to x.

Question 4: Differentiate f(x) = tan x with respect to x.

Question 5: Differentiate f(x) = (x² + 1)⁵ with respect to x

Question 6: Differentiate f(x) = √ x³ + 5 with respect to x.

Question 7: Differentiate f(x) = x²(x+5) with respect to x.

Question 8: Differentiate f(x) = x³ / 1 + x² with respect to x.

Question 9: Differentiate f(x) = sin x / x with respect to x.

Question 10: Differentiate f(x) = e^x / x + 4 with respect to x.

Question 11: Differentiate f(x) = Sin(3x²+2) with respect to x.

Question 12: Differentiate f(x) = ln (x² +1 ) with respect to x.

Answer Sheet

1) 7x⁶
2) 5
3) -sin x
4) sec² x
5) 10x( x² + 1) ⁴
6) 3x² / 2√x ³+5
7) 3x² + 10x
8) x⁴ + 3x²/ ( 1 + x ²) ²
9) x cos x -sinx / x ²
10) e^x (x + 3) / (x + 4) ²
11) 6x cos (3x² + 2)
12) 2x / x² +1

Check- Differentiation Quiz