A Step-by-Step Guide to Applying the Product Rule in Calculus for Finding Derivatives of Function Products

product rule

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

The product rule is a formula used in calculus to find the derivative of the product of two functions. It is used when you have two functions, let’s call them f(x) and g(x), and you want to find the derivative of their product, which is denoted as (f(x) * g(x)).

The formula for the product rule is as follows:

(d/dx)(f(x) * g(x)) = f'(x) * g(x) + f(x) * g'(x)

To apply the product rule, you need to differentiate each function separately, and then combine the results using the formula. Here’s a step-by-step guide on how to use the product rule:

1. Identify the two functions you want to find the derivative of their product, f(x) and g(x).

2. Differentiate f(x) to find f'(x), which represents the derivative of f(x) with respect to x.

3. Differentiate g(x) to find g'(x), which represents the derivative of g(x) with respect to x.

4. Apply the product rule formula:

(d/dx)(f(x) * g(x)) = f'(x) * g(x) + f(x) * g'(x)

5. Substitute the values of f'(x), g(x), f(x), and g'(x) into the equation.

6. Simplify the equation if possible.

Let’s see an example to illustrate the use of the product rule:

Find the derivative of the function h(x) = x^2 * sin(x)

1. Identify f(x) = x^2 and g(x) = sin(x).

2. Differentiate f(x):
f'(x) = 2x

3. Differentiate g(x):
g'(x) = cos(x)

4. Apply the product rule formula:
(d/dx)(x^2 * sin(x)) = 2x * sin(x) + x^2 * cos(x)

5. Substitute the values into the equation:
h'(x) = 2x * sin(x) + x^2 * cos(x)

6. Simplify the equation if possible, but in this case, there is no further simplification.

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

That’s the essence of the product rule. It allows you to find the derivative of the product of two functions.

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