# How do you differentiate e^((2-sqrtx)^2)  using the chain rule?

Nov 5, 2016

$y ' = {e}^{{\left(2 - \sqrt{x}\right)}^{2}} - \frac{2 {e}^{{\left(2 - \sqrt{x}\right)}^{2}}}{\sqrt{x}}$

#### Explanation:

This function can be simplified to ${e}^{4 - 4 \sqrt{x} + x}$.

We now use the rule that if $y = {e}^{f \left(x\right)} ,$ then $y ' = f ' \left(x\right) \times {e}^{f \left(x\right)}$.

So, letting $y = {e}^{\square}$, with $\square = 4 - 4 \sqrt{x} + x$, we will have to find the derivative of $\square$.

$\square ' = 1 - \left(0 \times 4 \sqrt{x} + \frac{1}{2 {x}^{\frac{1}{2}}} \times - 4\right) = 1 - \frac{2}{{x}^{\frac{1}{2}}}$

So, the derivative is $y ' = \left(1 - \frac{2}{\sqrt{x}}\right) {e}^{{\left(2 - \sqrt{x}\right)}^{2}} = {e}^{{\left(2 - \sqrt{x}\right)}^{2}} - \frac{2 {e}^{{\left(2 - \sqrt{x}\right)}^{2}}}{\sqrt{x}}$

Hopefully this helps!