Mastering the Product Rule: Calculus Basics for Finding Derivatives of Function Products

Product rule of f(x)g(x)

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

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

Let’s say we have 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 the following formula:

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

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

To use the product rule, you need to differentiate each function individually and then combine them using the given formula.

Let’s work through an example:

Example:
Consider the functions f(x) = x^2 and g(x) = sin(x).

1. Find the derivative of f(x):
To differentiate f(x) = x^2, we use the power rule. Taking the derivative of x^2 with respect to x gives us:
f'(x) = 2x

2. Find the derivative of g(x):
To differentiate g(x) = sin(x), we apply the derivative of the sine function. The derivative of sin(x) is:
g'(x) = cos(x)

3. Apply the product rule:
Now, we will apply the product rule to find the derivative of f(x) * g(x) = x^2 * sin(x).

Using the formula (f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x), we substitute in the derivatives we found in steps 1 and 2:
(f(x) * g(x))’ = (2x * sin(x)) + (x^2 * cos(x))

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

That’s how you apply the product rule to find the derivative of the product of two functions.

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