# If f(x)= sin6x  and g(x) = sqrt(x+3 , how do you differentiate f(g(x))  using the chain rule?

May 30, 2017

$\frac{d}{\mathrm{dx}} \left[f \left(g \left(x\right)\right)\right] = \frac{3 \cos \left(6 \sqrt{x + 3}\right)}{\sqrt{x + 3}}$

#### Explanation:

$f \left(x\right) = \sin \left(6 x\right)$
$g \left(x\right) = \sqrt{x + 3}$

$\therefore f \left(g \left(x\right)\right) = \sin \left(6 \sqrt{x + 3}\right)$

Applying the Chain Rule

$\frac{d}{\mathrm{dx}} \left[f \left(g \left(x\right)\right)\right] = \cos \left(6 \sqrt{x + 3}\right) \cdot \frac{d}{\mathrm{dx}} \left(6 \sqrt{x + 3}\right)$

$= \cos \left(6 \sqrt{x + 3}\right) \cdot \frac{d}{\mathrm{dx}} 6 {\left(x + 3\right)}^{\frac{1}{2}}$

Applying the Power Rule

$= \cos \left(6 \sqrt{x + 3}\right) \cdot \frac{6 {\left(x + 3\right)}^{- \frac{1}{2}}}{2}$

$= \frac{3 \cos \left(6 \sqrt{x + 3}\right)}{\sqrt{x + 3}}$