AcademyDifferential Calculus
Academy
Chain Rule
Level 1 - Calculus topic page in Differential Calculus.
Principle
The chain rule differentiates a composite function. It says that the rate of change of the outside function must be multiplied by the rate of change of the inside function.
Notation
\(f(g(x))\)
composition of an outer function f with inner function g
\(u\)
inner expression
\(\frac{du}{dx}\)
derivative of the inner expression
The Core Method
Identify the inner expression, differentiate the outside with respect to that inner expression, then multiply by the inner derivative.
Chain rule
\[\frac{d}{dx}f(g(x))=f'(g(x))g'(x)\]
For powers, if \(u=u(x)\), then \(\frac{d}{dx}u^n=nu^{n-1}u'\).
Worked Cases
Question
Differentiate \((3x^2+1)^5\).
Answer
Let \(u=3x^2+1\). Then the function is \(u^5\), whose derivative with respect to \(u\) is \(5u^4\). Also \(du/dx=6x\). Multiply: \(\frac{d}{dx}(3x^2+1)^5=5(3x^2+1)^4(6x)=30x(3x^2+1)^4\).
Examples
Question
Differentiate \(\sin(4x)\).
Answer
The outside derivative is \(\cos(4x)\), and the derivative of the inside \(4x\) is \(4\). Therefore \(\frac{d}{dx}\sin(4x)=4\cos(4x)\).