Practice Question on Differentiation Solved and Unsolved (Medium)

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

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Important Formulas of Differentiation

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

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Solved Practice Problems on Differentiation (Medium)

The following question focuses on differentiation at a Medium level.

Question 1: Differentiate t(x) = (ln x)⁵ with respect to x.

Solution:

Given, f (x) = (ln x)⁵

We have to use Chain rule to find derivative
Chain Rule = f'(g(x)).g'(x)
t'(x) = 5.(lnx)^5-1.1/x
t'(x) = 5 (ln x)⁴ .1/ x
t'(x) = 5 (ln x)⁴ / x

Question 2: Differentiate f(x) = x⁴e^x with respect to x.

Solution:

Given, f (x) = x⁴ e^x
We have to use product rule to find derivative

u = x⁴, u' = 4x³
v = e^x, v'= e^x

Product Rule = uv' + vu'
f'(x) = (x⁴) (e^x) + (e^x)(4x³)
f'(x) = x⁴e^x + 4x³e^x
f'(x) = e^x x³( x + 4)

Question 3: Differentiate f(x) = x⁵ ln (x) with respect to x.

Solution:

Given, f (x) = x⁵ ln x
We have to use product rule to find derivative
u = x⁵
u' = 5x⁴
v = ln x
v' = 1 / x

Product Rule = uv'+vu'
f'(x) = (x⁵)(1/x) + (ln x)(5x⁴)
f'(x) = x⁵/x + 5x⁴ ln x
f'(x) = x⁴ (1 +5 ln x)

Question 4: Differentiate f(x) = (x²+ 3x) sinx with respect to x.

Solution:

Given, f (x) = (x² + 3x).sinx

We have to use product rule to find derivative
u = x² + 3x, u' = 2x + 3
v = sinx, v'= cosx

Product Rule = uv'+vu'
f'(x) = (x²+3)(cosx) + (sinx)(2x + 3)
f'(x) = cosx(x ²+ 3) + sinx(2x + 3)

Question 5: Differentiate f(x) = x³+ 2x /e^x with respect to x.

Solution:

Given, f (x) = (x³ + 2x) / e^x

We have to use quotient rule to find derivative
u = x³+ 2x, u' = 3x² + 2
v = e ^x, v'= e ^x

Quotient Rule = u'v-uv'/v²
f'(x) = ( 3x² + 2 ) e^x - ( x³ + 2x ) e^x / (e^x)²
f'(x) = e^x ( 3x² + 2 - x³ + 2x) / ( e^x) ²
f'(x) = ( - x³ + 3x² - 2x + 2) / e^xthe

Question 6: The radius of a circular oil spill is increasing at a rate of 0.5 meters per second. How fast is the area of the spill increasing when the radius is 10 meters?

Solution:

Given,
Radius = 10 meter
Rate of change= 0.5 meter

area of circle = πr²
On differentiating w.r.t time,we get

dA/dt= 2 π (10) (0.5)
dA/dt= 10π

Therefore the area is increasing at rate of 10π meter per second.

Question 7: A balloon is inflating at 100 cm³/s. Find the rate of change of the radius when r = 5 cm.

Solution:

Given,
r = 5cm
dV/dt = 100 cm³/s

Volume of sphere = 4π r³/ 3
On differentiating w.r.t time,we get
dV / dt= 4 π r²dr/dt
100 = 4 π (5) ²dr/dt
100 = 100 π dr/dt
dr / dt = 100 / 100π
dr/dt = 1 / Therefore,rate of change in radius is 1 / π cm/s

Question 8: The temperature of a metal rod placed in the oven changes over time according to function: T(t) = (5t + 3)⁴ where t time is in minutes. Find the rate at which temperature is changing at any time t. Determine the rate of change when t = 2 minutes.

Solution:

Given,
The temprature of iron rod is given by a function = ( 5t + 3 ) ⁴
f(x) = ( 5t + 3 ) ⁴
f'(x) = 4 ( 5t + 3 ) ³
g(x) = 5t + 3
g'(x) = 5

Solving using Chain rule
T'(t) = f'(g(x)).g'(x)
T'(t) = 4(5t + 3) ³.5
T'(t) = 20 (5t + 3) ³

The general rate of temprature change at 20(5t + 3) ³

At t = 2 minutes
Substituting t = 2,
T'(2) = 20 (5(2) + 3) ³
T'(2) = 20(10 + 3) ³
T'(2) = 20( 13 )³
T'(2) = 43940

At t = 2 minutes, the temperature is increasing at a rate of 43,940°C per minute.

Unsolved Question on Differentiation

Question 1: Differentiate f(x) = x³ + 2 / x² + 1 with respect to x.

Question 2: Differentiate f(x) = x⁴ - 3x + 2/ tanx with respect to x.

Question 3: Differentiate f(x) = x²e^3x with respect to x.

Question 4: Differentiate f(x) = (x² + 5) ln x with respect to x.

Question 5: Differentiate f(x) = x³ sin( 2x + 1 ) with respect to x.

Question 6: A company sells the product at a price p(x) = 50 - 2x per unit, where x is the number of units sold. The total revenue is given by: R(x) = x p(x). Find the rate of change of revenue when 10 units are sold.

Question 7: A spherical balloon is being inflated so that the radius increases at a rate of 3 cm/sec. Find the rate at which volume is increasing when the radius is 10 cm.

Question 8: A square side is increasing at a rate of 2 cm/sec. Find the rate at which the area is increasing when the side length is 5cm.

Answer sheet

1) x⁴+ 3x² - 4x / (x² + 1)²
2) ((4x ³- 3) tanx - (x⁴ -3x + 2) sec²x )/ tan²x
3) e^3x x(2 + 3x)
4) 2x ln x + x + 5/x
5) 3x²sin(2x + 1) + 2x³cos(2x + 1)
6) 10
7) 1200 π cm³/sec
8) 20cm²/sec

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