How to Differentiate Quotient Functions Using the Quotient Rule | Math Tutorial and Example

Quotient Rule

The quotient rule is a formula used to differentiate functions that are expressed as the quotient of two functions

The quotient rule is a formula used to differentiate functions that are expressed as the quotient of two functions. It is derived from the product rule and is particularly useful when differentiating rational functions.

The quotient rule states that if we have a function y = f(x)/g(x), where both f(x) and g(x) are differentiable, then the derivative of y with respect to x is given by the following expression:

(dy/dx) = (g(x) * f'(x) – f(x) * g'(x))/[g(x)]^2

To apply the quotient rule, we follow these steps:

Step 1: Identify f(x) and g(x). These are the two functions that make up the quotient.

Step 2: Calculate the derivative of f(x) and g(x) separately. This step involves differentiating each function using any differentiation rules you have learned, such as the power rule, chain rule, etc.

Step 3: Plug the derivatives obtained in step 2 into the quotient rule formula. Specifically, substitute f'(x) for f'(x) and g'(x) for g'(x) in the numerator of the quotient rule formula.

Step 4: Simplify the expression obtained in step 3 to obtain the final derivative expression.

Let’s work through an example to illustrate how to use the quotient rule:

Example: Find the derivative of y = (5x^2 + 3)/(x^3 + 2)

Step 1: Identify f(x) and g(x). In our example, f(x) = 5x^2 + 3 and g(x) = x^3 + 2.

Step 2: Calculate f'(x) and g'(x).

f'(x) = d/dx(5x^2 + 3) = 10x

g'(x) = d/dx(x^3 + 2) = 3x^2

Step 3: Substitute f'(x) and g'(x) into the quotient rule formula.

(dy/dx) = [(x^3 + 2)(10x) – (5x^2 + 3)(3x^2)] / [(x^3 + 2)^2]

Step 4: Simplify the expression obtained in step 3.

(dy/dx) = (10x^4 + 20x – 15x^4 – 9x^2) / (x^6 + 4x^3 + 4)

(dy/dx) = (-5x^4 + 20x – 9x^2) / (x^6 + 4x^3 + 4)

So, the derivative of y = (5x^2 + 3)/(x^3 + 2) is (-5x^4 + 20x – 9x^2) / (x^6 + 4x^3 + 4).

This is the detailed process of applying the quotient rule to find the derivative of a quotient function.

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