# What is the derivative of arctan(6x)?

Aug 10, 2015

$\frac{6}{1 + 36 {x}^{2}}$

#### Explanation:

Recap that $\frac{d}{\mathrm{dx}} \arctan \left(x\right) = \frac{1}{1 + {x}^{2}}$

By the chain rule, if $y$ is a function of $u$ and $u$ is a function of $x$, then $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

Let $u = 6 x \setminus R i g h t a r r o w \frac{\mathrm{du}}{\mathrm{dx}} = 6$
$y = \arctan \left(6 x\right) = \arctan \left(u\right) \setminus R i g h t a r r o w \frac{\mathrm{dy}}{\mathrm{du}} = \frac{1}{1 + {u}^{2}}$

Therefore by the chain rule,
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$
$\frac{d}{\mathrm{dx}} \left(y\right) = \frac{1}{1 + {u}^{2}} \cdot 6$

Re-substituting $u = 6 x$ and $y = \arctan \left(u\right) = \arctan \left(6 x\right)$:
$\frac{d}{\mathrm{dx}} \arctan \left(6 x\right) = \frac{1}{1 + {\left(6 x\right)}^{2}} \cdot 6 = \frac{6}{1 + 36 {x}^{2}}$