Question

What is the difference between the chain rule and the power rule? Are they simply different forms of each other?

Answer 1

Hint: We first describe the basics of chain rule and the power rule with the use of general form. Then we use examples to understand the concept better.

Complete step by step solution:
The power rule is for only one function. We can take general form of power rule as $p\left( x \right)={{x}^{n}}$ and we get the derivative in the form of \[\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}\] .
On the other hand, chain rule is about how derivatives of composite functions work. We can take general form of chain rule as $f\left( x \right)=goh\left( x \right)$
Differentiating $f\left( x \right)=goh\left( x \right)$, we get
 \[\dfrac{d}{dx}\left[ f\left( x \right) \right] =\dfrac{d}{dx}\left[ goh\left( x \right) \right] =\dfrac{d}{d\left[ h\left( x \right) \right] }\left[ goh\left( x \right) \right] \times \dfrac{d\left[ h\left( x \right) \right] }{dx}={{g}^{'}}\left[ h\left( x \right) \right] {{h}^{'}}\left( x \right)\] .
Now we take examples to understand the concept better.
For power rule we take $p\left( x \right)={{x}^{3}}$ and we get the derivative as \[{{p}^{'}}\left( x \right)=\dfrac{d}{dx}\left( {{x}^{3}} \right)=3{{x}^{2}}\] .
For the chain rule we take a composite function where the main function is $g\left( x \right)=\sin x$ and the other function is $h\left( x \right)={{x}^{3}}$.
We have $goh\left( x \right)=g\left( {{x}^{3}} \right)=\sin {{x}^{3}}$. We take this as ours $f\left( x \right)=\sin \left( {{x}^{3}} \right)$.
The chain rule allows us to differentiate with respect to the function $h\left( x \right)$ instead of $x$ and after that we need to take the differentiated form of $h\left( x \right)$ with respect to $x$.
For the function $f\left( x \right)=\sin \left( {{x}^{3}} \right)$, we take differentiation of $f\left( x \right)=\sin \left( {{x}^{3}} \right)$ with respect to the function $h\left( x \right)={{x}^{3}}$ instead of $x$ and after that we need to take the differentiated form of $h\left( x \right)={{x}^{3}}$ with respect to $x$.
We know that differentiation of $g\left( x \right)=\sin x$ is ${{g}^{'}}\left( x \right)=\cos x$ and differentiation of $h\left( x \right)={{x}^{3}}$ is \[{{h}^{'}}\left( x \right)=3{{x}^{2}}\] . We apply the formula of \[\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}\] .
 \[\Rightarrow \dfrac{d}{dx}\left[ f\left( x \right) \right] =\dfrac{d}{d\left[ {{x}^{3}} \right] }\left[ \sin \left( {{x}^{3}} \right) \right] \times \dfrac{d\left[ {{x}^{3}} \right] }{dx}=3{{x}^{2}}\cos {{x}^{3}}\]
Therefore, differentiation of $\sin \left( {{x}^{3}} \right)$ is \[3{{x}^{2}}\cos {{x}^{3}}\] .
So, chain rule and the power rule aren’t simply different forms of each other.

Note: We need remember that in the chain rule \[\dfrac{d}{d\left[ h\left( x \right) \right] }\left[ goh\left( x \right) \right] \times \dfrac{d\left[ h\left( x \right) \right] }{dx}\] , we aren’t cancelling out the part \[d\left[ h\left( x \right) \right] \] . Canceling the base differentiation is never possible. It’s just a notation to understand the function which is used as a base to differentiate.