Let be a non-zero constant and define .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let

k

be a non-zero constant and define

g(x)=\sin(kx)

. (a) Formulate the difference quotient

\dfrac{g(x+h)-g(x)}{h}

and rewrite it as a single trigonometric expression involving

h

. Do not evaluate the limit yet. (b) Compute

\displaystyle \lim_{h\to 0}\dfrac{g(x+h)-g(x)}{h}

to obtain a formula for

g'(x)

in terms of

k

and

x

. Provide complete justification for each limit step. (c) Verify that your derivative formula reduces to the familiar numerical value when

k=1

and

x=\dfrac{\pi}{4}

. State the resulting slope.

Related theory

Derivative Foundations
Differentiation Rules
Differentiation from first principles
Derivatives
Limits and Derivatives

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