# How do you differentiate f(x)=e^((x^2+x^(1/2))^(1/2) ) using the chain rule?

Oct 27, 2016

For$x > 0 , f ' \left(x\right) = f \left(x\right) \frac{4 x \sqrt{x} + 1}{4 \sqrt{{x}^{2} \sqrt{x} + x}}$

#### Explanation:

For f(x) to be real and differentiable, x > 0.

Use $\left(f \left(x\right)\right) ' = \left({e}^{u}\right) ' u ' = f \left(x\right) u '$,

where u is the exponent function..-

So, $\left(f \left(x\right)\right) ' = f \left(x\right) \left(\sqrt{{x}^{2} + \sqrt{x}}\right) '$

$= f \left(x\right) \left(\frac{1}{2}\right) {\left({x}^{2} + \sqrt{x}\right)}^{- \frac{1}{2}} \left({x}^{2} + \sqrt{x}\right) '$

$= f \left(x\right) \frac{x + \frac{1}{4 \sqrt{x}}}{\sqrt{{x}^{2} + \sqrt{x}}}$

$= f \left(x\right) \frac{4 x \sqrt{x} + 1}{4 \sqrt{{x}^{2} \sqrt{x} + x}}$