Problem $4\mathrm{.}46.$ We will now prove Theorem $4\mathrm{.}44.$

$($a$)$ What is $m(xh)$ in terms of $f(xh)?($Your answer will also have $k$ in it$.)$

$($b$)$ What is $m(xh)-m(x)$ in terms of $f(xh)$ and $f(x)?($Your answer will also have $k$ in it$.)$

$($c$)$ Recall that the limit definition of $m^{\text{'}}(x)$ is

$m^{\text{'}}(x)=\underset{h0}{lim}\frac{m(xh)-m(x)}{h}$

Replace the numerator of this expression with your answer from part $($b$).$

$($d$)$ Rewrite your expression from part $($c$)$ by factoring out the constant $k$ from the numerator.

$($e$)$ Now rewrite your limit so the constant $k$ is in front of the limit$.$

$33$

$($f$)$ In your new expression, you have

$\underset{h0}{lim}\frac{f(xh)-f(x)}{h}$

What is this equal to$?$

$($g$)$ Finally, write $m^{\text{'}}(x)$ in terms of $f^{\text{'}}(x).($Your answer will also have $k$ in it$.)$