# What is the chain rule?

Nov 3, 2015

The chain rule for differentiation is essentially:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

So if $y = f \left(g \left(x\right)\right)$ then $\frac{d}{\mathrm{dx}} y = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

#### Explanation:

For example, suppose $y = {\left({x}^{2} + x - 1\right)}^{10}$

Let $u = {x}^{2} + x - 1$

Then $y = {u}^{10}$ and using the power rule:

$\frac{\mathrm{dy}}{\mathrm{du}} = 10 {u}^{9}$

$\frac{\mathrm{du}}{\mathrm{dx}} = 2 x + 1$

Hence:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}} = 10 {u}^{9} \cdot \left(2 x + 1\right) = 10 {\left({x}^{2} + x - 1\right)}^{9} \left(2 x + 1\right)$

Or we could formulate this as follows:

$g \left(x\right) = {x}^{2} + x - 1$

$f \left(u\right) = {u}^{10}$

$y = f \left(g \left(x\right)\right) = {\left({x}^{2} + x - 1\right)}^{10}$

$\frac{d}{\mathrm{dx}} y = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right) = 10 {\left(g \left(x\right)\right)}^{9} \cdot \left(2 x + 1\right)$

$= 10 {\left({x}^{2} + x + 1\right)}^{9} \left(2 x + 1\right)$