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### Section 3-10 : Implicit Differentiation

4. Find $$y'$$ by implicit differentiation for $$2{y^3} + 4{x^2} - y = {x^6}$$.

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Hint : Don’t forget that $$y$$ is really $$y\left( x \right)$$ and so we’ll need to use the Chain Rule when taking the derivative of terms involving $$y$$!
Start Solution

First, we just need to take the derivative of everything with respect to $$x$$ and we’ll need to recall that $$y$$ is really $$y\left( x \right)$$ and so we’ll need to use the Chain Rule when taking the derivative of terms involving $$y$$.

Differentiating with respect to $$x$$ gives,

$6{y^2}\,y' + 8x - y' = 6{x^5}$ Show Step 2

Finally, all we need to do is solve this for $$y'$$.

$\left( {6{y^2}\, - 1} \right)y' = 6{x^5} - 8x\hspace{0.5in} \Rightarrow \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{y' = \frac{{6{x^5} - 8x}}{{6{y^2}\, - 1}}}}$