I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 3.10 : Implicit Differentiation
5. Find \(y'\) by implicit differentiation for \(7{y^2} + \sin \left( {3x} \right) = 12 - {y^4}\).
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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 \(y'\).
\[\left( {14y + 4{y^3}} \right)y' = - 3\cos \left( {3x} \right)\hspace{0.5in} \Rightarrow \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{y' = \frac{{ - 3\cos \left( {3x} \right)}}{{14y + 4{y^3}}}}}\]