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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.
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 2Finally, all we need to do is solve this for \(y'\).
\[8x{y^7} - 5{x^4} - 2 = \left( {12{y^2} - 28{x^2}{y^6}} \right)y'\hspace{0.5in} \Rightarrow \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{y' = \frac{{8x{y^7} - 5{x^4} - 2}}{{12{y^2} - 28{x^2}{y^6}}}}}\]