Differentiation: Chain rule
The chain rule
We can also call a composition of functions a chain. The chain rule gives us a way to calculate the derivative of a composite function.
For a composite function #f(x)=\blue{g(}\green{h(x)}\blue{)}#, the following applies:
\[\begin{array}{c}
f'(x)=\orange{g'(\green{h(x)})}\cdot \purple{h'(x)}
\end{array}\]
f'(x) &=& \orange{4(\green{x^2-5x})^3} \cdot \purple{(2x-5)}
\end{array}\]
To use the chain rule, we can use this step-by-step guide.
Step-by-step guide chain rule |
Step-by-step |
Example |
Determine the derivative of a function that is composed of multiple functions: #f(x)=\blue{g(}\green{h(x)}\blue{)}#. |
#\qqquad \begin{array}{rcl} f(x)\phantom{'}&=&(x^2-1)^3\end{array}# |
|
Step 1 |
Distinguish the simpler functions #\blue{g(x)}# and #\green{h(x)}# of which #f(x)# is composed. |
#\qqquad\begin{array}{rcl}\blue{g(x)}\phantom{'}&=&\blue{x^3}\\ \green{h(x)}\phantom{'}&=&\green{x^2-1}\end{array}# |
Step 2 |
Determine the derivatives #\orange{g'(x)}# and #\purple{h'(x)}#. |
#\qqquad\begin{array}{rcl}\orange{g'(x)}&=&\orange{3x^2}\\ \purple{h'(x)}&=&\purple{2x}\end{array}# |
Step 3 |
Calculate the derivative of #f# with the formula: \[\begin{array}{c} |
#\qqquad \begin{array}{rcl} f'(x)&=& \orange{3(\green{x^2-1})^2}\cdot \purple{2x}\\&=&(3x^4-6x^2+3)\cdot \purple{2x}\\&=& 6x^5-12x^3+6x\end{array}# |
Step 1 | We distinguish the simpler functions #g(x)# and #h(x)# of which #f(x)# is composed. In other words, the functions for which #f(x)=g(h(x))#. #\begin{array}{rcl} g(x)&=&\displaystyle \sqrt{x}\\ h(x)&=& \displaystyle5-x^3\end{array}# |
Step 2 | We calculate the derivatives #g'(x)# and #h'(x)#. #\begin{array}{rcl} g'(x)&=& \displaystyle\dfrac{\dd}{\dd x}\left(\sqrt{x}\right)\\ &&\phantom{xxx}\blue{\text{definition derivative}}\\ &=& \displaystyle{{1}\over{2\cdot \sqrt{x}}}\\ &&\phantom{xxx}\blue{\text{power rule}}\end{array}# #\begin{array}{rcl} h'(x)&=& \displaystyle\dfrac{\dd}{\dd x}\left(5-x^3\right)\\ &&\phantom{xxx}\blue{\text{definition derivative}}\\ &=& \displaystyle-3\cdot x^2\\ &&\phantom{xxx}\blue{\text{sum rule and power rule}}\end{array}# |
Step 3 | We calculate the derivative #f'(x)#. #\begin{array}{rcl} f'\left(x\right)&=& \displaystyle g'(h(x))\cdot h'(x)\\ &&\phantom{xxx}\blue{\text{chain rule}}\\ &=& \displaystyle {{1}\over{2\cdot \sqrt{h\left(x\right)}}}\cdot -3\cdot x^2\\ &&\phantom{xxx}\blue{\text{substituting }g' \text{ and }h'}\\ &=& \displaystyle\displaystyle {{1}\over{2\cdot \sqrt{5-x^3}}} \cdot (-3\cdot x^2)\\ &&\phantom{xxx}\blue{h(x)\text{ substituted}}\\ &=& \displaystyle -{{3\cdot x^2}\over{2\cdot \sqrt{5-x^3}}}\\ &&\phantom{xxx}\blue{\text{simplified}} \end{array}# |
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