# How do you differentiate f(x)=tansqrtx using the chain rule?

Dec 26, 2015

You take it from the outside in. You can use the derivative of $\tan x$:

$\frac{d}{\mathrm{dx}} \left[\tan u\right] = {\sec}^{2} u \left(\frac{\mathrm{du}}{\mathrm{dx}}\right)$

where $u = \sqrt{x}$.

So:

$\frac{\mathrm{du}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} \left[\sqrt{x}\right] = \frac{1}{2} \cdot {x}^{\text{-1/2}} = \frac{1}{2 \sqrt{x}}$

As a result, you get:

$= \textcolor{b l u e}{\frac{{\sec}^{2} \sqrt{x}}{2 \sqrt{x}}}$