Solved Problems in Calculus

Introduction to solved problems in calculus:

Calculus is a major part of mathematics which deals with limits, differentiation, functions, derivatives, integrals, and infinite series. In modern mathematics, calculus has a major place and it was divided into two parts as differential calculus and integral calculus. Applications of calculus spread in science, economics, and engineering. It is insufficient to solve all the problems with algebra and calculus help us to solve all those problems.

Differentiation Using the Product Rule

Most of the calculus problems are solved by using the product rule. To solve the calculus problem, when derivative of a function f(x) is given, then it is equal to D{f(x)} or f'(x).
The product rule is used for differentiating problems where one function is multiplied by another. The rule follows the limit definition of derivative and it is given by,
D{f(x)g(x)} = f(x)g’(x) + f’(x)g(x)

Solved Problems in Calculus

Q 1 :     Differentiate, y = (x^3 + 7x - 1) (5x + 2)

Sol :       y’      = (x^3 + 7x - 1) D{5x + 2} + D{x^3 + 7x - 1} (5x + 2)

= (x^3 + 7x - 1) (5) + (3x^2 + 7) (5x + 2)

= 5x^3 + 35x – 5 + 15x^3 + 6x^2 + 35x + 14

= 20x^3 + 6x^2 + 70x + 9

Q 2 :  Differentiate, y = 5x^2 + sin x cos x

Sol :       y’      = 10x + sin x D{cos x} D{sin x} cos x

= 10x + sin x (-sin x) + (cos x) cos x

= 10x + cos2 x – sin^2 x

= 10x + cos (2x)

Q 3 : Differentiate, f(x) = (x + 8)4 sec (3x)

Sol :       f’(x)   = (x + 8)4 D{sec(3x)} + D{(x + 8)4} sec(3x)

= (x + 8)4 sec(3x) tan(3x) D{3x} + 4(x + 8)3 D(x + 8) sec(3x)

= (x + 8)4 sec(3x) tan(3x) (3) + 4(x+ 8)3 (1) sec(3x)

= (x + 8)3 sec(3x) {3(x + 8) tan(3x) + 4}

Q 4: Differentiate, y = x^2 sin3(5x).

Sol :       y’      = x^2 D{sin3(5x)} + D(x^2) sin3(5x)

= x^2 (3 sin^2(5x)) D{sin(5x)} + (2x) sin3(5x)

= 3x^2 sin^2(5x) cos(5x) D{5x} + 2x sin3(5x)

= 3x^2 sin^2(5x) cos(5x) (5) + 2x sin3(5x)

= 15x^2 sin^2 (5x) cos(5x) + 2x sin3 (5x)

= x sin^2(5x) {15x cos(5x) + 2 sin(5x)}