# How do you find the derivative of x^2 - y^2 = 16?

##### 1 Answer
May 26, 2015

I assume that we want to find $\frac{\mathrm{dy}}{\mathrm{dx}}$

${x}^{2} - {y}^{2} = 16$

Our choices are: solve for $y$ (make the function(s) explicit and differentiate

OR

leave the function $= f \left(x\right)$ implicit and use implicit differentiation.

Because this question was posted under Implicit Differentiation, we'll do that.

${x}^{2} - {y}^{2} = 16$
so the derivative with respect to $x$ of the left side equals the derivative with respect to $x$ of the right.

We write:

$\frac{d}{\mathrm{dx}} \left({x}^{2} - {y}^{2}\right) = \frac{d}{\mathrm{dx}} \left(16\right)$

Now find the derivatives (remembering that $y$ is some unknown function of $x$, so we need the chain rule to differentiate ${y}^{2}$

$2 x - 2 y \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

Solve for $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{x}{y}$