Given $f'(x)$ and $g'(x)$, what is $f''(2x)$?

Given $f'(x)$ and $g'(x)$, what is $f''(2x)$?

Question

The functions $f$, $g$ and $h$ are related by $f'(x) = g(x+1), \qquad g'(x)=h(x-1).$ It follows that $f''(2x)$ equals

1. $h(2x+1)$;

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1. $2h'(2x)$;

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1. $h(2x)$;

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1. $4h(2x)$.

Solution

Differentiating the first expression gives $f''(x) = g'(x+1).$ Then by replacing $x$ with $x+1$ in the second expression, we have $g'(x+1)=h(x)$ and so $f''(x)=h(x).$ Finally, if we replace $x$ with $2x$, we have $f''(2x)=h(2x)$ and so the answer is (c).

You might want to look at A tricky derivative for more on this problem.