Quotient Rule - Solved Examples

Example 1

Differentiate the following function using quotient rule of derivatives:

Apply the quotient rule. Multiply the denominator with the derivative of the numerator, and then subtract the product of the numerator and the derivative of the denominator. Divide the whole fraction by the square of the denominator.

 Derivative of sin(x) is cos(x)

 Take the derivative of 3x^2 + 1 by parts

 Derivative of 1 is 0. Factor out 3 from 3x^2 and apply power rule on x^2,

Example 2

Differentiate the following function using quotient rule:

Apply the quotient rule. Take the derivative of the numerator, multiply it with the denominator, then take the derivative of the denominator and multiply it with the numerator. Subtract the two and divide the fraction by the square of the denominator.

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

 Simplify the fraction by expanding the numerator and adding/subtracting like terms

Example 3

Use the quotient rule to find the derivative of the following function:

f(x) = 1/x

The numerator is 1 and denominator is x. Applying the quotient rule,  multiply the derivative of the numerator with the denominator and then multiply the derivative of the denominator with the numerator. Subtract the two. Divide by the square of the denominator

 Derivative of 1 is 0 and derivative of x is 1,

Example 4

Differentiate the following function by using the quotient rule:

The numerator is (10 – x) and the denominator is (x + 4). According to the quotient rule, multiply the derivative of the numerator with the denominator and then the derivative of the denominator with the numerator. Then subtract the second product from the first product.  Divide this by the square of the denominator. This gives the derivative of the quotient or the fraction.

Take the derivative of the sums term by term. Recall the rule that the derivative of a constant is zero. Derivative of x is 1 by power rule

 Simplify the expression