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

This question asks us to determine derivatives at x = 1 for composite functions built from f and g, namely h(x) = f(g(x)) and H(x) = f(g(x)).

First, recall the chain rule: if h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x). To evaluate h'(1), we need the values of g(1), f'(g(1)), and g'(1) from the given table. Similarly, since H is defined identically as H(x) = f(g(x)), we have H'(1) = h'(1) and thus H'(1) = f'(g(1)) · g'(1).

Here is what must be used and what is missing:
- We require g(1) to determine which f' is used in the product f'(g(1)). Without g(1), we do not know the correct input to f'.
- We require g'(1) to complete the product with f'(g(1)). Without g'(1), the derivative cannot be numerically determined.
- We also require the value of f'(at the specific argument g(1)) from the table; i.e., the derivative of f evaluated at the number g(1). If the table lists f' values directly for the corresponding input, we can take that value.

Because the answer field notes: 'Unable to answer. The table containing the values of f, f', g, and g' is missing.', it is clear that the necessary numerical data are not provided here. In a complete problem, you would:
- read off g(1) from the g column of the table,
- locate f'(g(1)) in the f' column at that same input,
- read g'(1) from the g' column at x = 1,
- multiply f'(g(1)) by g'(1) to obtain h'(1) and likewise H'(1).

If you later obtain the missing table values, apply the steps above in order and perform the multiplication to fill the blanks for (a) and (b). In short, with the data present, the computations are straightforward via the chain rule; with the data absent, no numeric result can be produced.

Any alternative approach would still require the same key inputs (g(1), g'(1), and f'(g(1))) from the provided table, so the limitation here is simply the missing data rather than a flaw in the method.

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