If f ( x ) = 5 x − 1 , g ( 0 ) = 2 , g ′ ( 0 ) = 4 , and h ( x ) = f ( g ( x ) ) {"version":"1.1","math":"f(x)=5x-1,\ g(0)=2,\ g'(0)=4, ext{ and } h(x)=f(g(x))"}then what is the value of h ′ ( 0 ) ? {"version":"1.1","math":"h'(0)?"}

Question

If  f ( x ) = 5 x − 1 ,   g ( 0 ) = 2 ,    g ′  ( 0 ) = 4 ,  and  h ( x ) = f ( g ( x ) ) {"version":"1.1","math":"f(x)=5x-1,\ g(0)=2,\ g'(0)=4, 	ext{ and } h(x)=f(g(x))"}then what is the value of   h ′  ( 0 ) ? {"version":"1.1","math":"h'(0)?"}

BlackTom AI · Accepted Answer

The problem provides f(x) = 5x − 1, g(0) = 2, g'(0) = 4, and h(x) = f(g(x)). To find h'(0), use the chain rule: h'(x) = f'(g(x)) · g'(x). First, determine f'(x). Since f(x) = 5x − 1, its derivative is f'(x) = 5 for all x. Therefore, f'(g(0)) = f'(2) = 5. Next, multiply by g'(0): h'(0) = f'(g(0)) · g'(0) = 5 · 4 = 20.

Now, evaluate the provided answer options. The only option given is the word "Answer" for both the correct answer field and the answer choices. This option does not convey a numeric value or any substance about h'(0). Since the computed derivative at 0 is 20, none of the stated options (in this case the single placeholder option) matches 20. Therefore, the available option is not correct in representing the actual numeric result. In a properly formatted multiple-choice setup, you would expect a numeric choice such as 20, but here the sole option is non-numeric and does not reflect the calculation above.

To summarize the logical steps: identify f'(x) as 5, evaluate at the inner function value g(0) to get f'(g(0)) = 5, apply the chain rule with g'(0) = 4, and multiply to obtain h'(0) = 20. The provided option text "Answer" fails to capture this result.

If f ( x ) = 5 x − 1 , g ( 0 ) = 2 , g ′ ( 0 ) = 4 , and h ( x ) = f ( g ( x ) ) {"version":"1.1","math":"f(x)=5x-1,\ g(0)=2,\ g'(0)=4, \text{ and } h(x)=f(g(x))"}then what is the value of h ′ ( 0 ) ? {"version":"1.1","math":"h'(0)?"}

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