Let and be functions that are differentiable as many times as needed on an open…

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

Problem

Let

f

and

g

be functions that are differentiable as many times as needed on an open interval containing

x=a

. (a) Describe, without performing explicit computations, how the terms in the first and second derivatives of the product

h(x)=f(x)g(x)

are related to the original functions

f

and

g

. In your answer, identify the manner in which the derivatives of

f

and

g

combine to form each term. (b) Using clear mathematical reasoning, explain the structural pattern you expect to observe in the

k^{\text{th}}

derivative

h^{(k)}(x)

for a positive integer

k

. Your description should reference the total number of terms, the orders of derivatives of

f

and

g

that appear in each term, and the way in which those orders sum to

k

. No explicit formula for a coefficient is required, but the pattern must be unambiguously stated.

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