The Product Rule: A Fundamental Tool in Calculus for Differentiating Functions Multiplied Together

Product Rule

The product rule is a method used to find the derivative of a product of two functions

The product rule is a method used to find the derivative of a product of two functions. It is a fundamental rule in calculus that helps us differentiate functions that are multiplied together.

The product rule states that if we have two functions, say f(x) and g(x), then their product, which we can represent as h(x) = f(x) * g(x), can be differentiated using the formula:

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

Let’s break down the formula and understand how it works:

– f'(x) represents the derivative of the first function, f(x).
– g(x) represents the second function as it is, without being differentiated.
– f(x) represents the first function as it is, without being differentiated.
– g'(x) represents the derivative of the second function, g(x).

To use the product rule, follow these steps:

Step 1: Identify f(x) and g(x) in the given problem.

Step 2: Differentiate f(x) to find f'(x).

Step 3: Differentiate g(x) to find g'(x).

Step 4: Plug the values into the product rule formula:

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

Step 5: Simplify the expression by multiplying and adding the terms accordingly.

Step 6: The result of this simplification is the derivative of the product function, h(x), with respect to x.

Using the product rule is essential when dealing with functions that are multiplied together, as it allows us to calculate the rate at which the product function is changing. This rule is an important tool in calculus to handle complex differentiation problems efficiently.

More Answers: