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

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

** 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

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

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

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

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

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

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

The given function is of the form U

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

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

Function f is of the form U

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

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

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

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

Function f is of the form U

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