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### Section 3-10 : Implicit Differentiation

12. Assume that $$x = x\left( t \right)$$, $$y = y\left( t \right)$$ and $$z = z\left( t \right)$$ and differentiate $${x^2} - {y^3} + {z^4} = 1$$ 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.
Show Solution

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.