The Product Rule | How to Find the Derivative of a Product Function in Calculus

Product Rule

The product rule is a formula used in calculus to find the derivative of a product of two functions

The product rule is a formula used in calculus to find the derivative of a product of two functions. Let’s say we have two functions, f(x) and g(x), and we want to find the derivative of their product:

(h(x) = f(x) * g(x))

The product rule states that the derivative of the product function, h'(x), is equal to:

h'(x) = f'(x) * g(x) + f(x) * g'(x)

In other words, to find the derivative of a product function, we differentiate one of the functions and multiply it by the other function as is, then add it to the differentiation of the other function multiplied by the first function as is.

Here’s an example to illustrate the product rule:

Let’s say we have the two functions:
f(x) = 3x² and g(x) = 2x³

To find the derivative of their product, h(x) = f(x) * g(x), we apply the product rule:

h'(x) = f'(x) * g(x) + f(x) * g'(x)

First, we find the derivatives of f(x) and g(x):
f'(x) = 6x (using the power rule for differentiation)
g'(x) = 6x² (using the power rule for differentiation)

Then, we substitute these derivatives into the product rule:
h'(x) = (6x) * (2x³) + (3x²) * (6x²)

Simplifying this expression, we have:
h'(x) = 12x⁴ + 18x⁴
h'(x) = 30x⁴

So, the derivative of the product function h(x) = f(x) * g(x) is h'(x) = 30x⁴.

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