Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 3.10 : Implicit Differentiation
5. Find \(y'\) by implicit differentiation for \(7{y^2} + \sin \left( {3x} \right) = 12 - {y^4}\).
Show All Steps Hide All Steps
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 \(y\)!
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,
\[14y\,y' + 3\cos \left( {3x} \right) = - 4{y^3}y'\] Show Step 2Finally, all we need to do is solve this for \(y'\).
\[\left( {14y + 4{y^3}} \right)y' = - 3\cos \left( {3x} \right)\hspace{0.5in} \Rightarrow \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{y' = \frac{{ - 3\cos \left( {3x} \right)}}{{14y + 4{y^3}}}}}\]