Calculus I - Related Rates

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Section 3.11 : Related Rates

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2. In the following assume that \(x\), \(y\) and \(z\) are all functions of \(t\). Given \(x = 4\), \(y = - 2\), \(z = 1\), \(x' = 9\) and \(y' = - 3\) determine \(z'\) for the following equation.

\[x\left( {1 - y} \right) + 5{z^3} = {y^2}{z^2} + {x^2} - 3\]

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Start Solution

The first thing that we need to do here is use implicit differentiation to differentiate the equation with respect to \(t\).

\[x'\left( {1 - y} \right) - x\,y' + 15{z^2}z' = 2y\,y'{z^2} + 2{y^2}z\,z' + 2x\,x'\] Show Step 2

All we need to do now is plug in the given information and solve for \(z'\).

\[27 + 12 + 15z' = 12 + 8\,z' + 72\hspace{0.5in} \Rightarrow \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{z' = {\textstyle{{45} \over 7}}}}\]