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

Nov 11, 2016

They are very different !

The "power rule" is used to differentiate a fixed power of $x$ e.g. ${x}^{3}$

The "chain rule" is used to differentiate a function of a function, e.g. ${e}^{\cos} x$, $\sin \left({x}^{3}\right)$, ${\left(1 + \ln x\right)}^{5}$ etc

#### Explanation:

Power Rule
$\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n} - 1$ where n' is a constant

Chain Rule

 d/dx(f(g(x) ) = f'(g(x)) * g'(x) # or $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$