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 function example 1Solution to Example 1:
Function f is the product of two functions: U = x 2 - 5 and V = x 3 - 2 x + 3; hence function example 1 written as a product of two functionsWe use the product rule to differentiate f as follows: derivative of a product of two functionswhere U ' and V ' are the derivatives of U and V respectively and are given by derivatives of functions U and VSubstitute to obtain derivatives of functions fExpand, group and simplify to obtain derivatives of functions f simplified




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 = 4x2 - 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
find derivative function example 3

Solution to Example 3:
The given function may be considered as the ratio of two functions: U = x2 + 1 and V = 5x - 3 and use the quotient rule to differentiate f is used as follows
derivative solution to example 3, step 1Expand and group to obtain f'(x) as follows
derivative solution to example 3, step 2




Example 4: Calculate the first derivative of function f given by
find derivative function example 4
Solution to Example 4:
Function f is the quotient of two functions hence the use of the quotient rule
derivative solution to example 4, step 1Write all terms in the numerator so that they have the same denominator 2 sqrt x
derivative solution to example 4, step 2Expand and group like terms to obtain f'(x)
derivative solution to example 4, step 3




Example 5: Calculate the first derivative of function f given by
find derivative function example 5
Solution to Example 5:
Function f given above may be considered as the product of functions U = 1/x - 3 and V = (x2 + 3)/(2x - 1), and function V may be considered as the quotient of two functions x2 + 3 and 2x - 1. We use the product rule for f and the quotient rule for V as follows
derivative solution to example 5, step 1Set all terms to a common denominator
derivative solution to example 5, step 2Expand and group to obtain the derivative f'.
derivative solution to example 5, step 3




Example 6: Calculate the first derivative of function f given by
find derivative function example 6
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)(x3 - x) and also consider V as the product of (2x - 1) and (x3 - x) and apply the product rule to f and V as follows
derivative solution to example 6, step 1Set a common denominator to all terms
derivative solution to example 6, step 2Expand and group like terms to obtain the derivative f'.
derivative solution to example 6, step 3




Example 7: Find the derivative of function f given by
find derivative function example 7
Solution to Example 7:
The given function is of the form U4 where U = x3 + 4. The use of the chain rule of differentiation gives f ' as follows
derivative solution to example 7, step 1Calculate U ' and substitute above to obtain f ' as follows
derivative solution to example 7, step 2




Example 8: Find the derivative of function f given by
find derivative function example 8
Solution to Example 8:
Function f is of the form U3 where U = (x - 1) / (x + 3). Apply the chain rule to obtain f ' as follows
derivative solution to example 8, step 1Calculate U ', using the quotient rule, and substitute to obtain
derivative solution to example 8, step 2Expand and group like terms to obtain a final form for the derivative f '
derivative solution to example 8, step 3




Example 9: Find the derivative of function f given by
find derivative function example 9
Solution to Example 9:
The given function is of the form sqrt U with U = x3 + 2 x + 1. Calculate U ' and use the chain rule to obtain
derivative solution to example 9




Example 10: Find the derivative of function f given by
find derivative function example 10
Solution to Example 10:
The given function is of the form U3/2 with U = x2 + 5. Apply the chain rule as follows
derivative solution to example 10, step 1Calculate U ', substitute and simplify to obtain the derivative f '.
derivative solution to example 10, step 2




Example 11: Find the derivative of function f given by
find derivative function example 11
Solution to Example 11:
Function f is of the form U1/4 with U = (x + 6)/(x + 5). Use the chain rule to calculate f ' as follows
derivative solution to example 11, step 1Since U is the quotient of two function, use the quotient rule to find U ' and substitute to obtain
derivative solution to example 11, step 2Expand and group like terms
derivative solution to example 11, step 3Change the negative exponent into a positive exponent to find a final form of f ' as follows
derivative solution to example 11, step 4

Exercises: Find the derivative of each of the following functions.
Find derivatives of functions

Answers to Above Exercises:
answers to above questions


More on
differentiation and derivatives
and also
Find Derivatives of Rational Functions - Calculators

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