A relation is given by for all points on a curve

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

A relation is given by

y^{x}=7

for all points

(x,y)

on a curve, with

y>0

. Students sometimes find it helpful to rewrite the relation before differentiating. Provide a clear, well-reasoned justification for the algebraic equivalence of the original equation and a rewritten form that expresses the product of

x

with a function of

y

equal to a constant. Then compare the differentiability of the two expressions and discuss why either form may be differentiated implicitly on the interval

y>0

. (a) Rewrite the relation in a form that involves the product

x\,g(y)=C

, where

g

is a differentiable function and

C

is a constant. Justify that this rewritten form is algebraically equivalent to

y^{x}=7

for

y>0

. (b) Explain why the rewritten form is differentiable with respect to

x

for

y>0

, and why differentiating either form implicitly must produce the same derivative

\dfrac{dy}{dx}

. Your explanation should reference relevant properties of differentiable functions and composition without explicitly calculating the derivative.

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