The graphs of and are often described as being phase–shifted versions of one another.

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The graphs of

y=\sin x

and

y=\cos x

are often described as being phase–shifted versions of one another. (a) Without performing a formal derivative calculation, explain why the instantaneous rate of change of

y=\sin x

x=0

equals the value of

y=\cos x

x=0

. Use either a unit–circle argument or a graphical argument in your explanation. (b) Using similar geometric or graphical reasoning, justify why the rate of change of

y=\cos x

is the negative of

y=\sin x

for all real numbers

x

. (c) Suppose the function is scaled vertically to

y=A\sin x

with

A>0

. Explain qualitatively why the amplitude of its derivative is

A

times larger than that of

\cos x

. Relate your explanation to the concept of rate of change.

Related theory

Differentiation Rules
Derivatives of standard functions via chain rule
Derivatives of composite functions
Continuity and Differentiability

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