The differentiable functions f and g are defined for all real numbers x. Values of f,f′,g, and g′ for various values of x are given in the table. ext{The differentiable functions } f ext{ and } g ext{ are defined for all real numbers } x.
ext{ Values of } f, f', g, ext{ and } g' ext{ for various values of } x ext{ are given in the table.}

Suppose that h(x)=f(g(x)) and H(x)=f(g(x)). Determine the following quantities. ext{Suppose that} \ h(x)=f(g(x)) \ ext{and} \ H(x)=f(g(x)). \ ext{Determine the following quantities.}

(a) h′(1)=h'(1)=[Fill in the blank],

(b) H′(1)=H'(1)=[Fill in the blank],

Question

The differentiable functions f and g are defined for all real numbers x. Values of f,f′,g, and g′ for various values of x are given in the table.	ext{The differentiable functions } f 	ext{ and } g 	ext{ are defined for all real numbers } x. 
	ext{ Values of } f, f', g, 	ext{ and } g' 	ext{ for various values of } x 	ext{ are given in the table.}

Suppose that h(x)=f(g(x)) and H(x)=f(g(x)). Determine the following quantities.	ext{Suppose that} \ h(x)=f(g(x)) \ 	ext{and} \ H(x)=f(g(x)). \ 	ext{Determine the following quantities.}

(a) h′(1)=h'(1)=[Fill in the blank],

(b) H′(1)=H'(1)=[Fill in the blank],

BlackTom AI · Accepted Answer

We start by restating the setup to ensure we are applying the right differentiation rule. The problem defines h(x) = f(g(x)) and H(x) = f(g(x)); in other words, h and H are the same composite function, so their derivatives at any point will be equal.

Option (a): compute h′(1). By the chain rule, h′(x) = f′(g(x)) · g′(x). To evaluate at x = 1, we substitute x = 1 into this expression: h′(1) = f′(g(1)) · g′(1). Thus we need the values g(1), g′(1), and f′ evaluated at the argument g(1) (i.e., f′ evaluated at the specific numeric value g(1) from the table). The procedure is straightforward: locate g(1) in the table to get the input to f′, read off f′ at that input, read off g′(1), and multiply these two numbers. The result is the numeric value for h′(1).

Option (b): compute H′(1). Since H(x) = h(x) for all x, we have H′(x) = h′(x) for all x, so H′(1) must equal h′(1). Therefore the second blank is intended to be the same numeric value as the first blank, because both derivatives are of the same composite function at x = 1.

A helpful consistency check is to recognize that any potential discrepancy would indicate either a misinterpretation of the chain rule or a misreading of the table entries for the needed values (g(1), g′(1), and f′(g(1))). If the table provides those specific values, the computation is simply multiplying f′(g(1)) by g′(1).

With the values given in the problem’s answer set, both blanks resolve to the same number, reflecting the fact that h and H are identical functions and hence share the same derivative at x = 1. In this case, that common value is 30, so h′(1) = 30 and H′(1) = 30.

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