Understanding the Product Rule in Calculus: A Complete Guide and Example

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 states that if you have two functions, let’s call them u(x) and v(x), their product is represented as u(x)v(x).

The formula for the product rule is:
(d/dx)[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

In words, to find the derivative of the product u(x)v(x), you first differentiate u(x) (call this u'(x)) while keeping v(x) as it is, and then you differentiate v(x) (call this v'(x)) while keeping u(x) as it is. Finally, you add the two resulting terms.

Let’s go through an example to illustrate the application of the product rule:

Example:
Find the derivative of f(x) = x² * sin(x)

Solution:
In this case, u(x) = x² and v(x) = sin(x).

Step 1:
Differentiate u(x):
u'(x) = 2x

Step 2:
Differentiate v(x):
v'(x) = cos(x)

Step 3:
Apply the product rule formula:
(d/dx)[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

(d/dx)[x² * sin(x)] = (2x * sin(x)) + (x² * cos(x))

Simplifying further, we get:
d/dx (x² * sin(x)) = 2x * sin(x) + x² * cos(x)

So, the derivative of f(x) = x² * sin(x) is 2x * sin(x) + x² * cos(x).

Note: The product rule is particularly useful when you have functions that cannot be easily differentiated without it. By following this rule, you can find the derivative of the product of any two functions.

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