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
(a) 19 (b) 9 (c) 17 (d) 14

Options

A. 19
B. 9
C. 17
D. 14

Explanation

`y=f(x)-f(2x) or y'=f'(x)-2f'(2x)`
`"or "y'(1)=f'(1)-2f'(2)=5, and " (1)"`
`y'(2)=f'(2)-2f'(4)=7" (2)"`
Now, let `y=f(x)-f(4x)`
`therefore" "y'=f'(x)-4f'(4x)`
`"or "y'(1)=f'(1)-4f'(4)" (3)"`
Substituting the value of `f'(2)=7+2f'(4) in (1),` we get
`f'(1)-2(7+2f'(4))=5`
`f'(1)-4f'(4)=19`