Answer
(a) $u'(1)$ = $0$
(b) $v'(5)$ = $-\frac{2}{3}$
Work Step by Step
(a)
From graphs of f and g
$f(1)$ = $2$
$g(1)$ = $1$
the slope of the line segment between (0,0) and(2,4) is $\frac{4-0}{2-0}$ = $2$
$f'(1)$ = $2$
the slope of the line segment between (-2,4) and(2,0) is $\frac{4-0}{-2-2}$ = $-1$
$g'(1)$ = $-1$
$u(x)$ = $f(x)g(x)$
$u'(x)$ = $f(x)g'(x)+g(x)f'(x)$
$u'(1)$ = $f(1)g'(1)+g(1)f'(1)$ = $2(-1)+1(2)$ = $0$
(b)
$v(x)$ = $\frac{f(x)}{g(x)}$
$v'(x)$ = $\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^{2}}$
$v'(5)$ = $\frac{g(5)f'(5)-f(5)g'(5)}{[g(5)]^{2}}$ = $\frac{2(-\frac{1}{3})-3(\frac{2}{3})}{2^{2}}$ = $-\frac{2}{3}$
