# How do you differentiate f(x)=e^(cossqrtx) using the chain rule.?

Jun 21, 2016

${e}^{\cos} \left(\sqrt{x}\right) \times \left(- \sin \left(\sqrt{x}\right)\right) \times \frac{1}{2 \sqrt{x}}$

#### Explanation:

Understand this that this is in the form of $f \left(g \left(h \left(x\right)\right)\right)$ where $f \left(x\right) = {e}^{x}$, $g \left(x\right) = \cos \left(x\right)$ and $h \left(x\right) = \sqrt{x}$

First, you need to know these things

$\frac{d}{\mathrm{dx}} \left({e}^{x}\right) = {e}^{x}$
$\frac{d}{\mathrm{dx}} \left(\cos \left(x\right)\right) = - \sin \left(x\right)$
$\frac{d}{\mathrm{dx}} \left(\sqrt{x}\right) = \frac{1}{2 \sqrt{x}}$

So, this becomes

Derivative = ${e}^{\cos} \left(\sqrt{x}\right) \times \left(- \sin \left(\sqrt{x}\right)\right) \times \frac{1}{2 \sqrt{x}}$