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### Section 3-10 : Implicit Differentiation

7. Find $$y'$$ by implicit differentiation for $$4{x^2}{y^7} - 2x = {x^5} + 4{y^3}$$.

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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 $$y$$! Also, don’t forget that because $$y$$ is really $$y\left( x \right)$$ we may well have a Product and/or a Quotient Rule buried in the problem.
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 $$y$$. This also means that the first term on the left side is really a product of functions of $$x$$ and hence we will need to use the Product Rule when differentiating that term.

Differentiating with respect to $$x$$ gives,

$8x{y^7} + 28{x^2}{y^6}y' - 2 = 5{x^4} + 12{y^2}y'$ Show Step 2

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

$8x{y^7} - 5{x^4} - 2 = \left( {12{y^2} - 28{x^2}{y^6}} \right)y'\hspace{0.5in} \Rightarrow \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{y' = \frac{{8x{y^7} - 5{x^4} - 2}}{{12{y^2} - 28{x^2}{y^6}}}}}$