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Section 3.10 : Implicit Differentiation
13. Assume that \(x = x\left( t \right)\), \(y = y\left( t \right)\) and \(z = z\left( t \right)\) and differentiate \({x^2}\cos \left( y \right) = \sin \left( {{y^3} + 4z} \right)\) with respect to \(t\).
Hint : This is just implicit differentiation like we’ve been doing to this point. The only difference is that now all the functions are functions of some fourth variable, \(t\). Outside of that there is nothing different between this and the previous problems.
Differentiating with respect to \(t\) gives,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{2x\,x'\cos \left( y \right) - {x^2}\sin \left( y \right)y' = \left( {3{y^2}y' + 4z'} \right)\cos \left( {{y^3} + 4z} \right)}}\]Note that because we were not asked to give the formula for a specific derivative we don’t need to go any farther. We could however, if asked, solved this for any of the three derivatives that are present.