For two functions \(f\) and \(g\) their composite, denoted \(f \circ g,\) is the function that results from applying \(g\) followed by \(f\). Symbolically, \({\bigl(f \circ {\color{DarkGreen} g}\bigr)(x) = f\bigl({\color{DarkGreen} g(x) }\bigr).}\) The chain rule is the name given to the formula for the derivative of the composite of two functions. \[ \frac{\mathrm{d}}{\mathrm{d}x}\biggl( f\bigl({\color{DarkGreen} g(x)}\bigr) \biggr) = \frac{\mathrm{d}}{\mathrm{d}x}f\bigl( {\color{DarkGreen} g(x)} \bigr) {\color{DarkGreen} \frac{\mathrm{d}}{\mathrm{d}x}g(x) } \qquad\text{ or }\qquad \Bigl(f\bigl({\color{DarkGreen} g(x)}\bigr)\Bigr)' = f'\bigl({\color{DarkGreen} g(x) }\bigr){\color{DarkGreen} g'(x) } \qquad\text{ or }\qquad \bigl(f \circ {\color{DarkGreen} g}\bigr)' = (f' \circ {\color{DarkGreen} g })({\color{DarkGreen} g' }) \] Take the derivative of the outer function \(f\) leaving the inner function \({\color{DarkGreen} g}\) the same, and then multiply by the derivative of the inner function \({\color{DarkGreen} g'}.\)
Examples
The chain rule can be applied iteratively to a composite of three or more functions, like un-nesting a set of matryoshka dolls. \[ \biggl(f\Bigl({\color{DarkGreen} g\bigl(}{\color{DarkGoldenRod} h(x)}{\color{DarkGreen}\bigr)}\Bigr)\biggr)' = f'\Bigl({\color{DarkGreen} g\bigl(}{\color{DarkGoldenRod} h(x)}{\color{DarkGreen}\bigr)}\Bigr) {\color{DarkGreen} g'\bigl(}{\color{DarkGoldenRod} h(x)}{\color{DarkGreen}\bigr)} {\color{DarkGoldenRod} h'(x)} \]