Mastering the Product Rule: A Comprehensive Guide to Calculus Derivatives of Product Functions

Product Rule

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

The product rule is a rule used in calculus to find the derivative of a product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.

Mathematically, if we have two functions f(x) and g(x), then the derivative of their product h(x) = f(x) * g(x) is given by:

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

Let’s work through an example to demonstrate how to use the product rule:

Suppose we want to find the derivative of the function h(x) = x^2 * sin(x). We have two functions here – f(x) = x^2 and g(x) = sin(x).

Step 1: Differentiate the first function f(x):
f'(x) = 2x

Step 2: Differentiate the second function g(x):
g'(x) = cos(x)

Step 3: Apply the product rule to find the derivative of the product h(x):
h'(x) = f'(x) * g(x) + f(x) * g'(x)
= (2x * sin(x)) + (x^2 * cos(x))

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

The product rule is a valuable tool in calculus and is often used when dealing with functions that involve multiplication. It allows us to find the derivative of such functions without having to resort to other, more complicated derivative rules.

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