Find the derivatives of various functions using different methods and rules in calculus. Several Examples with detailed solutions are presented. More exercises with answers are at the end of this page.

** Example 1:** Find the derivative of function f given by

f(x) = (x^2 - 5)(x^3 - 2x +3)

Function f is the product of two functions: U = x

f(x) = U V

f '(x) = U' V + U V'

U ' = 2x \; \text{and} \; V ' = 3x^2 - 2

f'(x) = 2 x(x^3- 2 x + 3) + (x^2 - 5) (3x^2 - 2)

= 2 x^4 - 4x^2 + 6x + 3x^4 - 2x^2 - 15x^2 + 10 \\
= 5x^4 -21x^2 + 10

** Example 2:** Calculate the first derivative of function f given by

f(x) = (\sqrt x + 2x)(4x^2-1)

This function may be considered as the product of function U = √x + 2x and V = 4x

f'(x) = U' V + U V' \\
= (\dfrac{1}{2\sqrt x} + 2)(4x^2-1) + (\sqrt x + 2 x)(8x)

f'(x) = \dfrac{(1+2\cdot2\sqrt x)(4x^2-1)+2\sqrt x(8x)(\sqrt x + 2x)}{2\sqrt x}

f'(x) = \dfrac{4x^2-1+16x^{5/2}-4\sqrt x+16x^2+32x^{5/2}}{2\sqrt x}

f'(x) = \dfrac{48x^{5/2}+20x^2-4x^{1/2}-1}{2\sqrt x}

** Example 3:** Calculate the first derivative of function f given by

__Solution to Example 3:__

The given function may be considered as the ratio of two functions: U = x^{2} + 1 and V = 5x - 3 and use the quotient rule to differentiate f is used as follows

** Example 4:** Calculate the first derivative of function f given by

Function f is the quotient of two functions hence the use of the quotient rule

** Example 5:** Calculate the first derivative of function f given by

Function f given above may be considered as the product of functions U = 1/x - 3 and V = (x

** Example 6:** Calculate the first derivative of function f given by

There are several ways to find the derivative of function f given above. One of them is to consider function f as the product of function U = sqrt x and V = (2x - 1)(x

** Example 7:** Find the derivative of function f given by

The given function is of the form U

** Example 8:** Find the derivative of function f given by

Function f is of the form U

** Example 9:** Find the derivative of function f given by

The given function is of the form sqrt U with U = x

** Example 10:** Find the derivative of function f given by

The given function is of the form U

** Example 11:** Find the derivative of function f given by

Function f is of the form U

__Exercises:__ Find the derivative of each of the following functions.

__Answers to Above Exercises:__

More on

differentiation and derivatives

and also

Find Derivatives of Rational Functions - Calculators