# How do you use implicit differentiation to find dy/dx given y^4=x^2-6x+2?

Sep 28, 2016

The answer is $\frac{2 x - 6}{4 {y}^{3}}$

#### Explanation:

So, the main thing to remember is that with implicit differentiation, when you take a derivative of something that's not an $x$, you need to do the following steps:

1. Take the derivative as normal.
2. Tag on a $\frac{\mathrm{dy}}{\mathrm{dx}}$

Then, to solve the problem, you'd just have to solve for the $\frac{\mathrm{dy}}{\mathrm{dx}}$

The only place this is really applicable in this particular example is with the ${y}^{4}$. So applying the above to this, we'd get $4 {y}^{3} \frac{\mathrm{dy}}{\mathrm{dx}}$.

We can find the derivatives for the other terms using simple power rule. This leaves us with:

$4 {y}^{3} \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x - 6$

Now, we just divide both sides by $4 {y}^{3}$ to solve for the $\frac{\mathrm{dy}}{\mathrm{dx}}$, and that leaves us with $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x - 6}{4 {y}^{3}}$, and we are done!

Hope that helps :)