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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 \(y'\).

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