Calculus I - Implicit Differentiation

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Section 3.10 : Implicit Differentiation

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12. Assume that \(x = x\left( t \right)\), \(y = y\left( t \right)\) and \(z = z\left( t \right)\) then differentiate \({x^2} - {y^3} + {z^4} = 1\) with respect to \(t\).

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Differentiating with respect to \(t\) gives,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{2x\,x' - 3{y^2}\,y' + 4{z^3}\,z' = 0}}\]

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.