The image you uploaded shows a problem that involves applying the chain rule for derivatives. It includes the graphs of two functions $f(x)$ and $g(x)$, and a few derivative problems based on these graphs. Let me break it down and provide solutions.

Problem Breakdown:

• Given Functions:

  • $g(x) = 2$, $f(a) = -3$ are some values provided.
  • You have graphs of $f(x)$ and $g(x)$ on the left side of the sheet, though their specific behavior needs to be analyzed visually to extract function values and slopes.

• Tasks:

  1. Chain Rule: $\frac{d}{dx}[g(f(x))] = g'(f(x)) f'(x)$ This is the formula needed for parts (a) and (b) under the chain rule, where you must apply the derivative to the composite function.

  2. Parts (a) and (b): Find $\frac{d}{dx}[g(f(x))]$ for specific values of $x$ (1 and -1).

  3. Part (d): Show $\frac{d}{dx} \left[ \frac{\sqrt{g(x)}}{f_2(x)} \right]$ at $x = 5$.

Part (a):

Find $\frac{d}{dx}[g(f(x))]$ at $x = 1$.

By applying the chain rule: $\frac{d}{dx}[g(f(x))] = g'(f(x)) f'(x)$

• Evaluate $f(1)$ from the graph.
• Find $f'(1)$, the slope of the tangent to the graph of $f(x)$ at $x = 1$.
• Use $g'(f(1))$, which depends on the value of $f(1)$.

Part (b):

Find $\frac{d}{dx}[g(f(x))]$ at $x = -1$.

Similarly, apply the chain rule: $\frac{d}{dx}[g(f(x))] = g'(f(x)) f'(x)$

• Evaluate $f(-1)$ and find $f'(-1)$ using the graph.
• Use $g'(f(-1))$ to complete the calculation.

Part (d):

Show that $\frac{d}{dx} \left[ \frac{\sqrt{g(x)}}{f_2(x)} \right] = \frac{\sqrt{2}}{8}$.

For this, apply the quotient rule combined with the chain rule.

• Use the formula for the derivative of a quotient: $\frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{v(x)u'(x) - u(x)v'(x)}{v(x)^2}$
• Set $u(x) = \sqrt{g(x)}$ and $v(x) = f_2(x)$.
• Find the derivatives $u'(x)$ and $v'(x)$, and plug them into the quotient rule formula.

Do you want me to walk through the solutions to specific parts of the problem in more detail?

Follow-up Questions:

1. How do we apply the chain rule to composite functions involving graphs?
2. How do we find $f'(x)$ and $g'(x)$ from a graph?
3. What is the difference between applying the chain rule and the quotient rule?
4. How can we interpret a derivative in terms of rates of change using graphical information?
5. What are the steps for solving a derivative problem with multiple rules (e.g., chain + quotient)?

Tip: When solving derivatives with both chain and quotient rules, always start by simplifying the expressions as much as possible before applying the rules.