Question

How to find h’(3) base on the graph

Answer 1

By the quotient rule for differentiation,

$\displaystyle h'(x) = \dfrac{g(x)f'(x) - f(x)g'(x)}{g(x)^2}$

According to the plots of $f$ and $g$, we have $f(3)=2$ and $g(3)=3$.

On the interval [2, 4], $f$ is a line through the points (2, 5) and (4, -1), and hence has slope (-1 - 5)/(4 - 2) = -3, so $f'(3)=-3$. We can similarly find $g'(3)=2$.

Then

$h'(3) = \dfrac{3\cdot(-3)-2\cdot2}{3^2} = \boxed{-\dfrac{13}9}$