Product Rule - Solved Examples

Example 1

Find the derivative of the following function using product rule:

 Multiply the first expression with the derivative of the second one, and then add this to the product of the second expression and the derivative of the first one.

 Take the derivative of each sum term by term. Factor out the constants from each term and then apply the power rule. If a term doesn’t have variables and is a constant, its derivative is zero

 Simplify the expression

Example 2

Differentiate the following function using product rule:

Convert each radical to a rational exponent,

Multiply the second multiplicand with the derivative of the first one, and then multiply the first multiplicand with the derivative of the second one.  Add the two products

 Apply power rule and take the derivative of each sum term by term,

 Simplify the expression by multiplying by expanding,

Example 3

Find the derivative of the following:

 Convert all radicals to rational exponents,

Apply the product rule,

 

Take the derivative of each sum one term by one term and apply power rule to each term,

Simplify the expression by expanding each product

Add and subtract like terms,

Example 4

Differentiate the following function by using the product rule and trigonometric derivative rules

 Applying product rule,

 Applying trigonometric derivative rules,

 Simplify the expression by multiplying and expanding, and then apply trigonometric formulas to simplify the expression

 Apply property sec(x) = 1/cos(x) and sin(x)/cos(x) = tan(x) and add/subtract like terms,