A differentiable function is defined on an open interval containing

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

Problem

A differentiable function

u

is defined on an open interval containing

x=\pi/4

, and

u(\pi/4)=1

. Consider the composite function

y=\cot\bigl(u(x)\bigr).

Respond to the prompts below. Express your answers in complete sentences supported by appropriate symbolic work. (a) Describe the sequence of differentiations required to obtain

\dfrac{dy}{dx}

in terms of

u(x)

and

u'(x)

. Identify the outermost and innermost functions involved in the composition, and explain the order in which their derivatives are taken. (b) Without computing a numerical value, write an explicit formula for

\dfrac{dy}{dx}

that involves only

u(x)

and

u'(x)

. Justify each factor in your formula by referencing the differentiation steps you described in part (a). (c) Suppose

u'(\pi/4)>0

. Explain how the sign of

\dfrac{dy}{dx}

x=\pi/4

depends on the sign of

\cot\bigl(u(\pi/4)\bigr)

and determine whether

y

is locally increasing or decreasing at

x=\pi/4

. Provide a brief verbal justification.

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