Limits, Continuity & Differentiability Question - None

Question

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:

Accepted Answer

Functional equations with derivatives
Let $g(x) = f(x) - f(2x)$. We have $g'(1) = 5$ and $g'(2) = 7$.
$g'(x) = f'(x) - 2f'(2x)$
At $x=1$: $f'(1) - 2f'(2) = 5$ ...(1)
At $x=2$: $f'(2) - 2f'(4) = 7$ ...(2)
Let $h(x) = f(x) - f(4x)$. Then:
$h'(x) = f'(x) - 4f'(4x)$
At $x=1$: $h'(1) = f'(1) - 4f'(4)$ ...(3)
From (2): $f'(4) = \frac{f'(2)-7}{2}$
Substituting: $h'(1) = f'(1) - 4\cdot\frac{f'(2)-7}{2} = f'(1) - 2f'(2) + 14 = 5 + 14 = \mathbf{19}$
Key concept: The derivative of $f(cx)$ is $c\cdot f'(cx)$; set up equations systematically.
Trap: Students try to find $f'(1)$, $f'(2)$, $f'(4)$ individually — unnecessary.

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