AcademyDifferentiation
Academy
Product Rule
Level 1 - Math I (Physics) topic page in Differentiation.
Product Rule
Used to differentiate products of functions.
Product Rule
\[\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\]
Memory Aid
"First times derivative of second plus second times derivative of first"
Example
Find \(\frac{d}{dx}[x^2 \sin(x)]\)
Let \(f(x) = x^2\), \(g(x) = \sin(x)\)
Example
\[\frac{d}{dx}[x^2 \sin(x)] = 2x \cdot \sin(x) + x^2 \cdot \cos(x)\]
Multiple Functions
For three functions:
Three Functions
\[\frac{d}{dx}[fgh] = f'gh + fg'h + fgh'\]
Leibniz Notation
Leibniz
\[\frac{d(uv)}{dx} = v\frac{du}{dx} + u\frac{dv}{dx}\]