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If f and g are the functions whose graphs are shown, let u(x)=f(g(x)),v(x)=g(f(x)), and w(x)=g(g(x)) Find each derivative, if it exists. If not explain why.
What does u'(1)=

To find \( u'(1) \), where \( u(x) = f(g(x)) \), we need to use the chain rule for differentiation. The chain rule states that if a function is composed of two functions, like \( u(x) = f(g(x)) \), then its derivative is given by:

\[
u'(x) = f'(g(x)) \cdot g'(x)
\]

Thus, to find \( u'(1) \), we will first need the following:
1. The value of \( g(1) \), which will tell us the input for \( f \) at that point.
2. The derivative \( f'(g(1)) \), which we can find from the slope of \( f(x) \) at \( g(1) \).
3. The derivative \( g'(1) \), which is the slope of the function \( g(x) \) at \( x = 1 \).

### Step-by-Step Process:
- **Find \( g(1) \)**:
  Looking at the graph, \( g(x) \) is the blue graph. From the image, we can see that \( g(1) = 4 \).

- **Find \( f'(4) \)**:
  Now, we need to find the slope of the function \( f(x) \) (the red graph) at \( x = 4 \). Observing the graph, \( f(x) \) is a constant between \( x = 3 \) and \( x = 5 \) with a value of approximately 2. Since it's a flat line, the derivative at this point is zero:
  \[
  f'(4) = 0
  \]

- **Find \( g'(1) \)**:
  Now, let's find the slope of \( g(x) \) at \( x = 1 \). From the graph, the blue line rises from \( x = 0 \) to \( x = 2 \) as \( y \) goes from 0 to 4. The slope can be calculated as:
  \[
  g'(1) = \frac{4 - 0}{2 - 0} = 2
  \]

### Apply the Chain Rule:
Now that we have all the components:
\[
u'(1) = f'(g(1)) \cdot g'(1) = f'(4) \cdot g'(1) = 0 \cdot 2 = 0
\]

Thus, \( u'(1) = 0 \).

Would you like a detailed explanation or more examples?

Here are 5 related questions to expand on this topic:
1. How would you compute \( v'(1) \) for \( v(x) = g(f(x)) \)?
2. How do you find \( w'(1) \) for \( w(x) = g(g(x)) \)?
3. What is the significance of the chain rule in multivariable functions?
4. Can you provide an example where the chain rule is applied to three functions?
5. How do you interpret a derivative graphically, especially for piecewise linear functions?

**Tip**: When using the chain rule, always check if the derivative of each function involved is defined at the points of interest, especially for piecewise or discontinuous functions.

If f and g are the functions whose graphs are shown, let u(x)=f(g(x)), v(x)=g(f(x)), and w(x)=g(g(x)). Find each derivative, if it exists. If not, explain why. What does u'(1)= ?

Find the derivative u'(1) using the chain rule for the composite function u(x) = f(g(x)). The solution involves evaluating the graphically represented functions f(x) and g(x), calculating slopes, and applying the chain rule for differentiation. Suitable for advanced high school or introductory calculus students.

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