Question

How to find h’(3)
base on the graph

Answer 1

By the quotient rule for differentiation,

`\displaystyle h'(x) = (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) = (3\cdot(-3)-2\cdot2)/(3^2) = \boxed{-\frac{13}9}`