https://file.aiquickdraw.com/mathbot/file/1728887717090we9t5az3.jpg

The image contains a mathematical problem written in Arabic, which translates to the following:

"If \( f(x) = (x+1)^{100} \), then find \( f'(1) \)."

To solve this, we need to differentiate \( f(x) \) with respect to \( x \), and then evaluate the derivative at \( x = 1 \).

### Step 1: Differentiate \( f(x) \)
Given:
\[
f(x) = (x+1)^{100}
\]

Using the power rule, \( \frac{d}{dx} (u^n) = n u^{n-1} \frac{du}{dx} \), where \( u = x+1 \) and \( n = 100 \), we get:
\[
f'(x) = 100 (x+1)^{99} \cdot \frac{d}{dx}(x+1) = 100 (x+1)^{99}
\]

### Step 2: Evaluate \( f'(1) \)
Substitute \( x = 1 \) into the derivative:
\[
f'(1) = 100 (1+1)^{99} = 100 \cdot 2^{99}
\]

Thus, \( f'(1) = 100 \cdot 2^{99} \).

Would you like more details or have any questions?

### Related Questions:
1. How would the solution change if \( f(x) = (x+2)^{50} \)?
2. What is the general derivative formula for \( f(x) = (ax+b)^n \)?
3. How can the power rule be extended to fractional exponents?
4. What are the implications of the power rule in solving real-world problems?
5. Can you find higher-order derivatives of \( f(x) = (x+1)^{100} \)?

### Tip:
Remember, the power rule simplifies differentiation by reducing it to a multiplication and subtraction step, making it useful for functions with large exponents.

Learn how to differentiate the function f(x) = (x+1)^100 and find f'(1) using the power rule. This calculus problem is suitable for students in Grades 11-12 and covers key concepts in differentiation.

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