https://file.aiquickdraw.com/mathbot/file/17294382820438k1g2p5b.jpg

The problem in the image asks to determine \( g'(1) \) given the following conditions:

- \( f(x) \) is a function such that \( f(1) = 13 \) and \( f'(1) = 11 \).
- The function \( g(x) = x^5 \cdot f(x) \).

We are asked to find \( g'(1) \).

### Step-by-step solution:
To solve for \( g'(x) \), we'll use the **product rule** of differentiation, since \( g(x) = x^5 \cdot f(x) \) is a product of two functions: \( x^5 \) and \( f(x) \).

The product rule states:
\[
g'(x) = \frac{d}{dx}(x^5) \cdot f(x) + x^5 \cdot \frac{d}{dx}(f(x))
\]

Let's differentiate each term:

1. The derivative of \( x^5 \) is:
   \[
   \frac{d}{dx}(x^5) = 5x^4
   \]

2. The derivative of \( f(x) \) is \( f'(x) \).

Now, applying the product rule:
\[
g'(x) = 5x^4 \cdot f(x) + x^5 \cdot f'(x)
\]

To find \( g'(1) \), substitute \( x = 1 \), \( f(1) = 13 \), and \( f'(1) = 11 \):
\[
g'(1) = 5(1)^4 \cdot f(1) + (1)^5 \cdot f'(1)
\]
\[
g'(1) = 5 \cdot 13 + 1 \cdot 11
\]
\[
g'(1) = 65 + 11 = 76
\]

Thus, the value of \( g'(1) \) is \( 76 \).

Would you like more details on this process or have any questions?

Here are 5 related questions to expand your understanding:
1. How does the product rule of differentiation work with more complex functions?
2. How would the solution change if \( f'(1) \) were different?
3. Can you derive the product rule from first principles using limits?
4. What happens if \( g(x) \) involves three functions multiplied together?
5. How is the chain rule applied when differentiating a composition of functions?

**Tip:** When using the product rule, ensure you carefully differentiate each component before applying the rule, and don’t forget to substitute values if evaluating at a specific point.

Misalkan fungsi f di x = 1 dengan f(1) = 13, dan f'(1) = 11. Jika g(x) = x^5 · f(x), Tentukan g'(1) = ______.

Learn how to differentiate g(x) = x^5 · f(x) using the product rule and find g'(1) with f(1) = 13 and f'(1) = 11. This step-by-step solution covers key calculus concepts, including the product rule and differentiation of power functions. Suitable for students in Grades 11-12.

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