We are given the composite function $u(x) = f(g(x))$, and the task is to find $u'(6)$. This requires applying the chain rule for differentiation, which states:

$u'(x) = f'(g(x)) \cdot g'(x)$

We need to evaluate this at $x = 6$, so:

$u'(6) = f'(g(6)) \cdot g'(6)$

Step 1: Find $g(6)$

From the graph of $g(x)$ (the blue graph), when $x = 6$, we see that $g(6) = 3$.

Step 2: Find $g'(6)$

Next, we look for the derivative of $g(x)$ at $x = 6$, which corresponds to the slope of the function at that point. From the graph, $g(x)$ is a horizontal line at $x = 6$, so the slope (derivative) is:

$g'(6) = 0$

Step 3: Find $f'(g(6)) = f'(3)$

Since $g(6) = 3$, we now need to find $f'(3)$, which is the derivative of $f(x)$ at $x = 3$. From the graph of $f(x)$ (the red line), the slope between $x = 3$ and $x = 4$ is $-1$ (a straight line descending with a slope of $-1$).

Thus:

$f'(3) = -1$

Step 4: Calculate $u'(6)$

Now, applying the chain rule:

$u'(6) = f'(g(6)) \cdot g'(6) = f'(3) \cdot g'(6) = (-1) \cdot 0 = 0$

So, $u'(6) = 0$.

Would you like more details or have any questions?

Related Questions:

1. How do you interpret the slope of a graph geometrically?
2. Can the chain rule be applied to more than two functions?
3. What happens when one of the derivatives in the chain rule is zero?
4. What other methods can be used to find a derivative if the graph is not provided?
5. How would the result change if $g'(6)$ were not zero?

Tip: When working with composite functions, always ensure you apply the chain rule correctly, focusing on the inner and outer functions' derivatives separately.