# How do you differentiate y=x*sqrt(16-x^2)?

Sep 26, 2015

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{2} / \sqrt{16 - {x}^{2}} + \sqrt{16 - {x}^{2}}$

#### Explanation:

This is a product of 2 functions so we use the product rule which states that the derivative of the product of 2 functions is the first function times the derivative of the second, plus the second function times the derivative of the first.

Inside this product rule we will also require the power rule which states that the derivative of a function to a power is the power times the function raised to 1 less than the given power, multiplied by the derivative of the function.

Putting this all together, we get :

$\frac{\mathrm{dy}}{\mathrm{dx}} = x \cdot \frac{1}{2} {\left(16 - {x}^{2}\right)}^{- \frac{1}{2}} \cdot 2 x + {\left(16 - {x}^{2}\right)}^{\frac{1}{2}} \cdot \left(1\right)$

$= {x}^{2} / \sqrt{16 - {x}^{2}} + \sqrt{16 - {x}^{2}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{2} / \sqrt{16 - {x}^{2}} + \sqrt{16 - {x}^{2}} = {x}^{2} / \sqrt{16 - {x}^{2}} + \frac{16 - {x}^{2}}{\sqrt{16 - {x}^{2}}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{16}{\sqrt{16 - {x}^{2}}}$