# Find Derivatives of Functions in Calculus

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

__Solution to Example 1:__Function f is the product of two functions: U = x

^{ 2}- 5 and V = x

^{ 3}- 2 x + 3; hence We use the product rule to differentiate f as follows: where U ' and V ' are the derivatives of U and V respectively and are given by Substitute to obtain Expand, group and simplify to obtain

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

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

__Solution to Example 2:__This function may be considered as the product of function U = √x + 2x and V = 4x

^{2}- 1, hence the use of the product rule

\[ f'(x) = U' V + U V' \\ = (\dfrac{1}{2\sqrt x} + 2)(4x^2-1) + (\sqrt x + 2 x)(8x) \] To add the above you need to write all terms as fractions with a common denominator.

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

\[ f'(x) = \dfrac{4x^2-1+16x^{5/2}-4\sqrt x+16x^2+32x^{5/2}}{2\sqrt x} \] and group to obtain the final result for the derivative of f as follows.

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

** 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

Expand and group to obtain f'(x) as follows

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

__Solution to Example 4:__Function f is the quotient of two functions hence the use of the quotient rule

Write all terms in the numerator so that they have the same denominator 2 sqrt x

Expand and group like terms to obtain f'(x)

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

__Solution to Example 5:__Function f given above may be considered as the product of functions U = 1/x - 3 and V = (x

^{2}+ 3)/(2x - 1), and function V may be considered as the quotient of two functions x

^{2}+ 3 and 2x - 1. We use the product rule for f and the quotient rule for V as follows

Set all terms to a common denominator

Expand and group to obtain the derivative f'.

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

__Solution to Example 6:__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

^{3}- x) and also consider V as the product of (2x - 1) and (x

^{3}- x) and apply the product rule to f and V as follows

Set a common denominator to all terms

Expand and group like terms to obtain the derivative f'.

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

__Solution to Example 7:__The given function is of the form U

^{4}where U = x

^{3}+ 4. The use of the chain rule of differentiation gives f ' as follows

Calculate U ' and substitute above to obtain f ' as follows

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

__Solution to Example 8:__Function f is of the form U

^{3}where U = (x - 1) / (x + 3). Apply the chain rule to obtain f ' as follows

Calculate U ', using the quotient rule, and substitute to obtain

Expand and group like terms to obtain a final form for the derivative f '

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

__Solution to Example 9:__The given function is of the form sqrt U with U = x

^{3}+ 2 x + 1. Calculate U ' and use the chain rule to obtain

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

__Solution to Example 10:__The given function is of the form U

^{3/2}with U = x

^{2}+ 5. Apply the chain rule as follows

Calculate U ', substitute and simplify to obtain the derivative f '.

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

__Solution to Example 11:__Function f is of the form U

^{1/4}with U = (x + 6)/(x + 5). Use the chain rule to calculate f ' as follows

Since U is the quotient of two function, use the quotient rule to find U ' and substitute to obtain

Expand and group like terms

Change the negative exponent into a positive exponent to find a final form of f ' as follows

__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