$[4/6$ Points$]$

SCALCET$93\mathrm{.}4\mathrm{.}071.$

If $f$ and $g$ are the functions whose graphs are shown, let $u(x)=f(g(x)),v(x)=g(f(x)),$ and $w(x)=g(g(x)).$ Find each derivative, if it exists. If it does not exist, explain why. $($If an answer does

$($a$)u^{\text{'}}(1)=\frac{1}{5}$

It does exist.

$u^{\text{'}}(1)$ does not exist because $f^{\text{'}}(5)$ does not exist.

$u^{\text{'}}(1)$ does not exist because $g^{\text{'}}(1)$ does not exist.

$u^{\text{'}}(1)$ does not exist because $f^{\text{'}}(6)$ does not exist.

$u^{\text{'}}(1)$ does not exist because $g^{\text{'}}(5)$ does not exist.

$($b$)$ It does exist.

$v^{\text{'}}(1)$ does not exist because $f^{\text{'}}(1)$ does not exist.

$v^{\text{'}}(1)$ does not exist because $g^{\text{'}}(1)$ does not exist.

$v^{\text{'}}(1)$ does not exist because $f^{\text{'}}(2)$ does not exist.

$v^{\text{'}}(1)$ does not exist because $g^{\text{'}}(2)$ does not exist.

$($c$)w^{\text{'}}(1)=\frac{1}{4}x$

It does exist.

$w^{\text{'}}(1)$ does not exist because $f^{\text{'}}(1)$ does not exist.

$w^{\text{'}}(1)$ does not exist because $g^{\text{'}}(1)$ does not exist.

$w^{\text{'}}(1)$ does not exist because $f^{\text{'}}(5)$ does not exist. $w^{\text{'}}(1)$ does not exist because $g^{\text{'}}(5)$ does not exist.