# How do you implicitly differentiate 15=x^2-y^2/x?

Mar 4, 2016

See the explanation section, below.

#### Explanation:

First of all, I would rewrite the equation. (Why use the quotient rule when we don't need to?)

$15 x = {x}^{3} - {y}^{2}$ $\text{ }$ (for $x \ne 0$)

Now differentiate:

$\frac{d}{\mathrm{dx}} \left(15 x\right) = \frac{d}{\mathrm{dx}} \left({x}^{3}\right) - \frac{d}{\mathrm{dx}} \left({y}^{2}\right)$

$15 = 3 {x}^{2} - 2 y \frac{\mathrm{dy}}{\mathrm{dx}}$

Solve for $\frac{\mathrm{dy}}{\mathrm{dx}}$

$2 y \frac{\mathrm{dy}}{\mathrm{dx}} = 3 {x}^{2} - 15$

So, $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 {x}^{2} - 15}{2 y}$