# Question #6afc0

Apr 12, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{e}^{8} \left(1 + 6 {x}^{5}\right)}{2 \sqrt{x + {x}^{6}}}$

#### Explanation:

$y = {e}^{8} \sqrt{x + {x}^{6}}$

First of all, note that ${e}^{8}$ is a (multiplicative) constant. It will remain on the outside of the function as we differentiate it.

For the $\sqrt{x + {x}^{6}}$ portion, we should rewrite it as ${\left(x + {x}^{6}\right)}^{\frac{1}{2}}$ and then differentiate it using the product and chain rules.

Since the derivative of ${x}^{\frac{1}{2}}$ is $\frac{1}{2} {x}^{- \frac{1}{2}}$, the derivative of $f {\left(x\right)}^{\frac{1}{2}}$ is $\frac{1}{2} f {\left(x\right)}^{- \frac{1}{2}} \cdot f ' \left(x\right)$.

Thus, the derivative of the function is:

$\frac{\mathrm{dy}}{\mathrm{dx}} = {e}^{8} \left(\frac{1}{2} {\left(x + {x}^{6}\right)}^{- \frac{1}{2}}\right) \cdot \frac{d}{\mathrm{dx}} \left(x + {x}^{6}\right)$

The derivative of $x + {x}^{6}$ is $1 + 6 {x}^{5}$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{e}^{8} \left(1 + 6 {x}^{5}\right)}{2 \sqrt{x + {x}^{6}}}$