Consider the following,
(a) Find by implieit differentiation.
(b) Solve the equation explicitly for and differentiate to get in terms of .
© Check that your solutions to parts (a) and (b) are consistent by substituting the expression for into your solution for part (a).

To find by implicit differentiation, let’s start by assuming we have an equation involving and , such as . Using implicit differentiation, we differentiate both sides with respect to , remembering to apply the chain rule for . For instance, if we had , differentiating gives , leading to .

Now, if we explicitly solve for , rearranging the equation could yield . Differentiating directly gives . Finally, substituting this expression for back into the implicit differentiation solution will confirm that both methods yield consistent derivatives, validating our work.

So, both parts (a) and (b) give you and , ensuring they match when we compute it correctly.