Understanding the Product Rule in Calculus: How to Find the Derivative of a Product of Two Functions

Product Rule

The product rule is a rule in calculus that allows you to find the derivative of the product of two functions

The product rule is a rule in calculus that allows you to find the derivative of the product of two functions. It is stated as follows:

If f(x) and g(x) are two differentiable functions, then the derivative of their product, denoted as (f(x) * g(x))’, is given by:

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

In other words, to find the derivative of a product of two functions, you take the derivative of the first function and multiply it by the second function, then add it to the product of the first function with the derivative of the second function.

Let’s go through an example to demonstrate how to apply the product rule:

Example:
Find the derivative of f(x) = x^2 * sin(x)

Solution:
To apply the product rule, we need to identify the two functions that are being multiplied together. In this case, the first function is f(x) = x^2 and the second function is g(x) = sin(x).

Now, let’s find the derivatives of the individual functions:
– f'(x) = 2x (using the power rule for differentiation)
– g'(x) = cos(x) (using the derivative of sin(x) which is cos(x))

Now, we can apply the product rule:
(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)
= (2x * sin(x)) + (x^2 * cos(x))

So, the derivative of f(x) = x^2 * sin(x) is (2x * sin(x)) + (x^2 * cos(x)).

Remember to simplify your answer if possible.

The product rule is extremely helpful when dealing with functions that are multiplied together. It allows us to find the derivative of the product without having to differentiate each term separately.

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