The Product Rule | Differentiating the Product of Two Functions in Calculus

Product Rule

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

The product rule is a formula used in calculus to differentiate the product of two functions. It is a rule that allows us to find the derivative of a product by taking the derivative of each function and then combining them.

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 is given by:

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

In other words, to find the derivative of f(x) * g(x), we differentiate f(x) and multiply it by g(x), and then add to it the product of f(x) and the derivative of g(x).

Here’s a step-by-step breakdown of how to apply the product rule:

1. Differentiate f(x) to find f'(x), the derivative of f(x).
2. Differentiate g(x) to find g'(x), the derivative of g(x).
3. Multiply f'(x) by g(x) to get the first term of the derivative.
4. Multiply f(x) by g'(x) to get the second term of the derivative.
5. Add the two terms obtained in steps 3 and 4 to get the final derivative, h'(x).

It’s important to note that the product rule only applies when differentiating the product of two functions. If you have more than two functions being multiplied together, you’ll need to use the product rule multiple times or consider using other differentiation techniques.

The product rule is a fundamental concept in calculus and is often used in problems involving functions that are products of multiple terms. It allows us to find the rate of change or slope of a function that is the product of two other functions.

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