Problems and Solutions

Go through the below calculus problems to understand the process of differentiation and integration.

Problem 1: Let f(y) = e^y and g(y) = 10y. Use the chain rule to calculate h′(y) where h(y) = f(g(y)).

Solution: Given,

f(y) = e^y and

g(y) = 10y

First derivative above functions are

f'(y) = e^y and

g'(y) = 10

To find: h′(y)

Now, h(y) = f(g(y))

h'(y) = f'(g(y))g'(y)

h'(y) = f'(10y)10

By substituting the values.

h'(y) = e^10y x 10

or h'(y) = 10 e^10y

Problem 2: Integrate sin 3x + 2x with respect to x.

Solution: Given instructions can be written as:

∫ sin 3x + 2x dx

Use the sum rule, which implies

∫  sin 3x dx+ ∫ 2x dx ……… Equation 1

Solve ∫  sin 3x dx first.

use substitution method,

let 3x = u => 3 dx = du (after derivation)

or dx = 1/3 du

=> ∫  sin 3x dx turned as∫ sin u  X 1/3 du

or 1/3 ∫  sin u du

which is 1/3 (-cos u) + C, where C= constant of integration

Substituting values again, we get

∫  sin 3x dx= -cos(3x)/3 + C ……… Equation 2

Solve∫ 2x dx

∫ 2x dx = 2∫  x dx = 2 * x²/2 + C = x² + C ……. Equation 3

Equation (1) => ∫  sin 3x dx+ ∫ 2x dx

= -cos(3x)/3 + x² + C