Product Rule Guide

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Learn how to differentiate the product of two functions.

Step-by-Step Logic

Set u = f(x) and v = g(x)
Find u' and v'
Plug into uv' + vu'

Worked Example

Problem: Find derivative of x² * sin(x)

u = x² ⇒ u' = 2x
v = sin(x) ⇒ v' = cos(x)
Apply Form: (x²)(cos(x)) + (sin(x))(2x)
Result: x²cos(x) + 2xsin(x)

About this calculator

What is the Product Rule?

The product rule is a formula used to find the derivative of a function that is the product of two or more functions. It states that the derivative of a product is the first function times the derivative of the second, plus the second function times the derivative of the first.

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Pro Tips

Identify two functions, u and v, such that your function is y = u * v.
Calculate u' and v' separately first.
Combine them using the formula: uv' + vu'.

Fun Facts

"The product rule is a special case of the general Leibniz rule for higher-order derivatives."

Formula: d/dx [f(x)g(x)] = f(x)g'(x) + g(x)f'(x)