Question

Suppose the function $f(x) - f(2x)$ has the derivative 5 at $x = 1$ and derivative '7' at $x = 2$, then the derivative of $f(x)-f(4x)-10x$ at $x = 1$ is equal to

Answer 1

9

Answer 2

Let $g(x) = f(x) - f(2x)$. We are given $g'(1) = 5$ and $g'(2) = 7$. The derivative of $g(x)$ is $g'(x) = f'(x) - 2f'(2x)$. So, $f'(1) - 2f'(2) = 5$ (1) and $f'(2) - 2f'(4) = 7$ (2).

Let $h(x) = f(x) - f(4x) - 10x$. We want to find $h'(1)$. The derivative of $h(x)$ is $h'(x) = f'(x) - 4f'(4x) - 10$. So, $h'(1) = f'(1) - 4f'(4) - 10$.

From (1), $f'(1) = 5 + 2f'(2)$. From (2), $2f'(4) = f'(2) - 7$, so $4f'(4) = 2(f'(2) - 7) = 2f'(2) - 14$.

Substitute these into the expression for $h'(1)$: $h'(1) = (5 + 2f'(2)) - (2f'(2) - 14) - 10$ $h'(1) = 5 + 2f'(2) - 2f'(2) + 14 - 10$ $h'(1) = 5 + 14 - 10 = 9$.