Suppose the function f(x) - f(2x) has the der | vedclass

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(A) Let $g(x) = f(x) - f(2x)$. Then $g'(x) = f'(x) - 2f'(2x)$.Given $g'(1) = 5$,we have $f'(1) - 2f'(2) = 5$  --- $(1)$.Given $g'(2) = 7$,we have $f'(

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$19$

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(A) Let $g(x) = f(x) - f(2x)$. Then $g'(x) = f'(x) - 2f'(2x)$.Given $g'(1) = 5$,we have $f'(1) - 2f'(2) = 5$  --- $(1)$.Given $g'(2) = 7$,we have $f'(2) - 2f'(4) = 7$  --- $(2)$.We want to find the derivative of $h(x) = f(x) - f(4x)$ at $x = 1$.$h'(x) = f'(x) - 4f'(4x)$.At $x = 1$,$h'(1) = f'(1) - 4f'(4)$.From $(2)$,$f'(2) = 7 + 2f'(4)$.Substitute $f'(2)$ into $(1)$:$f'(1) - 2(7 + 2f'(4)) = 5$$f'(1) - 14 - 4f'(4) = 5$$f'(1) - 4f'(4) = 19$.Thus,$h'(1) = 19$.

Suppose the function $f(x) - f(2x)$ has the derivative $5$ at $x = 1$ and derivative $7$ at $x = 2$. The derivative of the function $f(x) - f(4x)$ at $x = 1$ has the value equal to:

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