Questions: Fill in the Blank 3 points The graphs of f and g are given below. Compute the derivatives. If an answer does not exist, type DNE. (d/dx) f(g(x))(x=0)= type your answer.. (d/dx) f(g(x))(x=6)= type your answer..

Solution Steps

We have to evaluate $\frac{d}{dx}f(g(x))|_{x=0}$. By the chain rule, we know that $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$.

$g(0)=0$.

So, $f'(g(0)) = f'(0)$. The graph of $f$ passes through the points $(0,9)$ and $(4,0)$. Since this segment is a line, we calculate the slope as $m = \frac{0-9}{4-0} = \frac{-9}{4}$. Therefore, $f'(0) = -\frac{9}{4}$.

The graph of g passes through (0,0) and (6,10). Since this is a straight line, the slope is $m = \frac{10-0}{6-0} = \frac{10}{6} = \frac{5}{3}$. Therefore, $g'(0)=\frac{5}{3}$.

$\frac{d}{dx}f(g(x))|_{x=0} = f'(g(0))g'(0) = f'(0)g'(0) = -\frac{9}{4} \cdot \frac{5}{3} = -\frac{45}{12} = -\frac{15}{4}$.

$\frac{d}{dx}f(g(x))|_{x=6} = f'(g(6))g'(6)$. $g(6) = 10$. $f'(10)$ can be calculated by considering the line segment between $(4,0)$ and $(10,4)$. The slope is $\frac{4-0}{10-4} = \frac{4}{6} = \frac{2}{3}$. So $f'(10)=\frac{2}{3}$.

Next, we notice that at $x=6$, $g'(x)$ is passing through the points $(6,10)$ and $(12,4)$. Since it's a line segment, we find the slope to be $\frac{4-10}{12-6} = \frac{-6}{6} = -1$.

So, $g'(6)=-1$. Then $\frac{d}{dx}f(g(x))|_{x=6} = f'(g(6))g'(6) = f'(10)g'(6) = \frac{2}{3} \cdot (-1) = -\frac{2}{3}$.

$\frac{d}{dx}g(f(x))|_{x=4} = g'(f(4))f'(4)$. From the graph, we know that $f(4)=0$, so $g'(f(4)) = g'(0) = \frac{5}{3}$. $f'(4)$ lies on the line segment between $(4,0)$ and $(10,4)$, so $f'(4) = \frac{4-0}{10-4} = \frac{4}{6} = \frac{2}{3}$. Therefore, $\frac{d}{dx}g(f(x))|_{x=4} = g'(0)f'(4) = \frac{5}{3} \cdot \frac{2}{3} = \frac{10}{9}$.

Final Answer: