Consider the implicit relation x y^ 2 + y=4

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Consider the implicit relation

x y^{2}+\cos y=4.

The graph of this relation defines

y

as an implicit function of

x

near most points on the curve. (a) Differentiate the relation implicitly with respect to

x

and write the resulting equation that involves

\frac{dy}{dx}

. Do not solve for

\frac{dy}{dx}

in this step. (b) Algebraically isolate

\frac{dy}{dx}

to obtain an explicit formula in terms of

x

and

y

. Explain every transformation you apply while solving for

\frac{dy}{dx}

. In your explanation, indicate how the presence of

y

as a function of

x

affects the differentiation of each term. (c) In complete sentences, describe why

\frac{dy}{dx}

appears in more than one term of the differentiated equation and discuss the interplay between multiplications of

x

and

y

and compositions involving

y

that leads to this structure.