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

$y=f\left(x\right)-f(2 x)$

$y^{\text{'}}=f^{\text{'}}\left(x\right)-2f^{\text{'}}\left(2x\right)$

$y\text{'}\left(1\right)=f\text{'}\left(1\right)-2f\text{'}\left(2\right)=5\text{, and }y\text{'}\left(2\right)=f\text{'}\left(2\right)-2f\text{'}\left(4\right)=7$

$y=f\left(x\right)-f(4 x)$

$y^{\text{'}}=f^{\text{'}}\left(x\right)-4f^{\text{'}}\left(4x\right)$

$\Rightarrow y^{\text{'}}\left(1\right)=f^{\text{'}}\left(1\right)-4f^{\text{'}}\left(4\right)$

$\text{ Substituting the value of }{f}^{\text{'}}\left(2\right)=7+2f^{\text{'}}\left(4\right)\text{ in Eq. (i), }$

$f^{\text{'}}\left(1\right)-2\left[7+2f^{\text{'}}\left(4\right)\right]=5$

$\Rightarrow f^{\text{'}}\left(1\right)-4f^{\text{'}}\left(4\right)=19$