A calculus student is examining how the numerical coefficient and the exponent of a…

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

Problem

A calculus student is examining how the numerical coefficient and the exponent of a single-term function affect the form of its derivative. The student records three prototype functions in the form

f(x)=k\,x^{n}

, where

k

and

n

are real numbers and

x>0

. (a) Describe in words how changing the value of

k

influences the derivative of

f

. Your description should address both magnitude and sign, and it should be valid for any real exponent

n

. Cite one example to support your explanation. (b) Describe in words how changing the value of

n

influences the derivative of

f

. Specifically discuss the role of

n

in determining both the new exponent of

x

that appears after differentiation and the multiplicative factor in front of that power. Cite one example to support your explanation. (c) The student claims that replacing

k

with

-k

f(x)

always multiplies

f'(x)

-1

. Determine whether the claim is true for all real

n

and justify your conclusion algebraically. (d) Give an explicit example of a function

f(x)=k\,x^{n}

with

n<0

and verify that your explanations in parts (a) and (b) remain valid for your chosen negative exponent.

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