Donna Flynn

## Answered question

2022-07-23

I'm stuck on this problem involving implicit differentiation.
The instructions ask me to find ${y}^{\prime }$, the problem is:
$\left(x-2y{\right)}^{3}=2{y}^{2}-3$
So far I've been able to get this far:
$3\left(x-2y{\right)}^{2}\left(1-2{y}^{\prime }\right)=4y\left({y}^{\prime }\right)$
I've been trying to manipulate it for a while but I can't figure out how to finish the problem properly.

### Answer & Explanation

Monica Dennis

Beginner2022-07-24Added 13 answers

Another way could be to consider the implicit function
$f=\left(x-2y{\right)}^{3}-2{y}^{2}+3=0$
Compute its derivative with respect to $x$ and $y$
${f}_{x}^{\prime }=3\left(x-2y{\right)}^{2}$
${f}_{y}^{\prime }=-6\left(x-2y{\right)}^{2}-4y$
Now, use the implicit function theorem
$\frac{dy}{dx}=-\frac{{f}_{x}^{\prime }}{{f}_{y}^{\prime }}=-\frac{3\left(x-2y{\right)}^{2}}{-6\left(x-2y{\right)}^{2}-4y}=\frac{3}{2}×\frac{\left(x-2y{\right)}^{2}}{3\left(x-2y{\right)}^{2}+2y}$

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