Power Rule Implies Chain Rule

Having developed the Product Rule d\left( uv \right)=u{v}'+{u}'v and the Power Rule \frac{d}{dx}{{x}^{n}}=n{{x}^{n-1}} for derivatives in your class, you can explore similar rules for the product of more than two functions and suddenly the Chain Rule will appear.

For three functions use the associative property of multiplication with the rule above:

d\left( uvw \right)=d\left( \left( uv \right)w \right)=u\cdot v\cdot dw+w\cdot d(uv)=u\cdot v\cdot dw+w\left( udv+vdu \right)

So expanding with a slight change in notation:

d\left( uvw \right)=uv{w}'+u{v}'w+u'vw

For four factors there is a similar result:

d\left( uvwz \right)=uvw{z}'+uv{w}'z+u{v}'wz+{u}'vwz

Exercise: Let {{f}_{i}} for i=1,2,3,...,n be functions. Write a general formula for the derivative of the product {{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}} as above and in sigma notation


d\left( {{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}} \right)={{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{{f}'}_{n}}+{{f}_{1}}{{f}_{2}}{{{f}'}_{3}}\cdots {{f}_{n}}+{{f}_{1}}{{{f}'}_{2}}{{f}_{3}}\cdots {{f}_{n}}+\cdots +{{{f}'}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}}

\displaystyle d\left( {{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}} \right)=\sum\limits_{i=1}^{n}{\frac{{{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}}}{{{f}_{i}}}{{{{f}'}}_{i}}}

This idea may now be used  to see the Chain Rule appear. Students may guess that d{{\left( f \right)}^{4}}=4{{\left( f \right)}^{3}}, but wait there is more to it.

Write {{\left( f \right)}^{4}}=f\cdot f\cdot f\cdot f\text{ }. Then from above

d{{\left( f \right)}^{4}}=d\left( f\cdot f\cdot f\cdot f\text{ } \right)=f\cdot f\cdot f\cdot {f}'+f\cdot f\cdot {f}'\cdot f+f\cdot {f}'\cdot f\cdot f+{f}'\cdot f\cdot f\cdot f

d{{\left( f \right)}^{4}}=4{{\left( f \right)}^{3}}{f}'\text{ }

Looks just like the power rule, but there’s that “extra” {f}'. Now you are ready to explain about the Chain Rule in the next class.


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