2. Suppose the function $f(x)-f(2 x)$ has derivative 5 at $x=1$ and derivative 7 at $x=2$. Find the derivative of $f(x)-f(4 x)$ at $x=1$.
Solution:
19

Let $g(x)=f(x)-f(2 x)$. Then we want the derivative of
$$
f(x)-f(4 x)=(f(x)-f(2 x))+(f(2 x)-f(4 x))=g(x)+g(2 x)
$$
at $x=1$. This is $g^{\prime}(x)+2 g^{\prime}(2 x)$ at $x=1$, or $5+2 \cdot 7=19$.