The image provided asks to solve two problems based on the given information $f(x) = x^3 + 5x$ and $g(x) = x$. The problems require finding $f'(7) \cdot g'(7)$ and $f'(7) \cdot g(7)$.

Step 1: Find the derivatives of $f(x)$ and $g(x)$

1. For $f(x) = x^3 + 5x$: $f'(x) = 3x^2 + 5$

2. For $g(x) = x$: $g'(x) = 1$

Step 2: Evaluate the derivatives and functions at $x = 7$

1. Evaluate $f'(7)$: $f'(7) = 3(7)^2 + 5 = 3(49) + 5 = 147 + 5 = 152$

2. Evaluate $g'(7)$: $g'(7) = 1$

3. Evaluate $g(7)$: $g(7) = 7$

Step 3: Calculate the required values

1. For $f'(7) \cdot g'(7)$: $f'(7) \cdot g'(7) = 152 \cdot 1 = 152$

2. For $f'(7) \cdot g(7)$: $f'(7) \cdot g(7) = 152 \cdot 7 = 1064$

Final Answers:

1. $f'(7) \cdot g'(7) = 152$
2. $f'(7) \cdot g(7) = 1064$

Would you like further details or explanations on any part of this problem?

Here are 5 related questions to expand your understanding:

1. How do you find the derivative of a composite function like $h(x) = f(g(x))$?
2. What is the difference between a function and its derivative?
3. How can you apply the product rule when finding the derivative of the product of two functions?
4. How do you interpret the physical meaning of a derivative in real-world applications?
5. What happens if you evaluate the derivative of a function at its critical points?

Tip: Always check if functions are differentiable at the point of interest before proceeding with derivative calculations!