Applying the Product Rule to Find the Derivative of a Product of Two Functions

derivative of product

The derivative of a product of two functions can be found using the product rule

The derivative of a product of two functions can be found using the product rule.

Let’s consider two functions: f(x) and g(x). The product rule states that the derivative of the product of these two functions, denoted as (f(x) * g(x)), is given by:

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

Here, f'(x) represents the derivative of the function f(x) with respect to x, and g'(x) represents the derivative of the function g(x) with respect to x.

To apply the product rule, you would first take the derivative of the first function, f'(x), and multiply it by the second function, g(x). Then, you would add the product of the first function, f(x), and the derivative of the second function, g'(x).

Let’s see an example to better understand the process.

Example:
Let’s find the derivative of the function f(x) = x^2 * sin(x).

In this case, the two functions are f(x) = x^2 and g(x) = sin(x).

Using the product rule, we have:

(f(x) * g(x))’ = (x^2 * sin(x))’ = (2x * sin(x)) + (x^2 * cos(x))

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

It’s important to note that the product rule can be applied to more than two functions as well. For example, if we have three functions, f(x), g(x), and h(x), the product rule can be applied in a similar manner:

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

Remember to carefully apply the product rule whenever you need to find the derivative of a product of functions.

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