Answer
(a) $u'(1)$ = $\frac{3}{4}$
(b) $v'(x)$ does not exist
(c) $w'(1)$ = $-2$
Work Step by Step
(a)
$u(x)$ = $f(g(x))$
$u'(x)$ = $f'(g(x))g'(x)$
so
$u'(1)$ = $f'(g(1))g'(1)$
$u'(1)$ = $f'(3)g'(1)$
find $f'(3)$
from (2,4) to (6,3)
$m$ = $\frac{3-4}{6-2}$ = $-\frac{1}{4}$
so $f'(3)$ = $-\frac{1}{4}$
find $g'(1)$
from (0,6) to (2,0)
$m$ = $\frac{0-6}{2-2}$ = $-3$
so $g'(1)$ = $-3$
Thus
$u'(1)$ = $f'(3)g'(1)$
$u'(1)$ = $-\frac{1}{4}(-3)$
$u'(1)$ = $\frac{3}{4}$
(b)
$v(x)$ = $g(f(x))$
$v'(x)$ = $g'(f(x))f'(x)$
$v'(1)$ = $g'(f(1))f'(1)$
$v'(1)$ = $g'(2)f'(1)$
which does not exist since $g'(2)$ does not exist
(c)
$w(x)$ = $g(g(x))$
$w'(x)$ = $g'(g(x))g'(x)$
$w'(1)$ = $g'(g(1))g'(1)$
$w'(1)$ = $g'(3)g'(1)$
find $g'(3)$
from (2,0) to (5,2)
$m$ = $\frac{2-0}{5-2}$ = $\frac{2}{3}$
so $g'(1)$ = $\frac{2}{3}$
Thus
$w'(1)$ = $\frac{2}{3}(-3)$ = $-2$
