Derivatives · Definition
Chain rule
Before you start you need to know
The chain rule lets you differentiate compositions of functions, one of the most used tools in calculus.
Definition
The chain rule gives the derivative of a composition \(f(g(x))\): the outer derivative evaluated at the inner function, times the derivative of the inner function.
When one function sits inside another, \(f(g(x))\), changes chain together: if \(g\) changes \(g'\) times as fast as \(x\) and \(f\) changes \(f'\) times as fast as \(g\), the total effect is the product. Hence \((f\circ g)'=f'(g(x))\,g'(x)\).
It is the rule most often forgotten and most often used: almost every real function is a composition, such as \(\sin(3x)\), \(e^{x^{2}}\) or \((x^{2}+1)^{5}\).
Practical recipe: differentiate "the outside" leaving the inside untouched, then multiply by the derivative of "the inside".
Think about it
Differentiate \((x^2+1)^5\).
\(5(x^2+1)^4\cdot 2x=10x(x^2+1)^4\).
Exercises 10
Differentiate the following functions
- \(\displaystyle f(x)=\sin {x^{2}}\)
- \(\displaystyle f(x)=\left(3 x + 1\right)^{5}\)
- \(\displaystyle f(x)=e^{x^{2}}\)
- \(\displaystyle f(x)=\ln{\left(x^{2} + 1 \right)}\)
- \(\displaystyle f(x)=\sqrt{1 - x^{2}}\)
- \(\displaystyle f(x)=\left(x^{2} + x\right)^{3}\)
- \(\displaystyle f(x)=\cos(\ln(x^{2}+1))\)
- \(\displaystyle f(x)=\sqrt{\sin x}\)
Problems
References
- James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
- Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.