# How do you find dy/dx given x=3y^(1/3)+2y?

Jan 24, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = {y}^{\frac{2}{3}} / \left(1 + 2 {y}^{\frac{2}{3}}\right)$

#### Explanation:

We are differentiating with respect to $x$. This means that $\frac{d}{\mathrm{dx}} x = 1$, but $\frac{d}{\mathrm{dx}} y = \frac{\mathrm{dy}}{\mathrm{dx}}$.

This will be recurrent throughout this problem: differentiating anything with $y$, since it's a function that's not $x$, will take the chain rule.

Thus, the derivative with respect to $x$ of ${y}^{2}$ is not just $2 y$, but $2 y \frac{\mathrm{dy}}{\mathrm{dx}}$.

Now proceeding my differentiating:

$\frac{d}{\mathrm{dx}} x = 3 \frac{d}{\mathrm{dx}} {y}^{\frac{1}{3}} + 2 \frac{d}{\mathrm{dx}} y$

$1 = {y}^{- \frac{2}{3}} \frac{\mathrm{dy}}{\mathrm{dx}} + 2 \frac{\mathrm{dy}}{\mathrm{dx}}$

Factoring:

$\frac{\mathrm{dy}}{\mathrm{dx}} \left({y}^{- \frac{2}{3}} + 2\right) = 1$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{{y}^{- \frac{2}{3}} + 2}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = {y}^{\frac{2}{3}} / \left(1 + 2 {y}^{\frac{2}{3}}\right)$