https://file.aiquickdraw.com/mathbot/file/1727111221036iz1kxzbt.jpg

Let's start by solving problem 11, which states:

**Problem 11**:
Let \( f(x) = e^{g(x)} \), where \( g(0) = 6 \) and \( g'(0) = 2 \). Find \( f'(0) \).

### Solution:
To find \( f'(x) \), we need to differentiate \( f(x) = e^{g(x)} \) using the chain rule.

1. The derivative of \( f(x) = e^{g(x)} \) is:
   \[
   f'(x) = e^{g(x)} \cdot g'(x)
   \]
2. Now, substitute \( x = 0 \):
   \[
   f'(0) = e^{g(0)} \cdot g'(0)
   \]
3. We're given that \( g(0) = 6 \) and \( g'(0) = 2 \), so:
   \[
   f'(0) = e^6 \cdot 2
   \]
   \[
   f'(0) = 2e^6
   \]

Thus, the value of \( f'(0) \) is \( 2e^6 \).

Would you like more details or any further explanation on this solution?

---

Here are five related questions for further practice:

1. If \( h(x) = \ln(g(x)) \) and \( g(0) = 6 \), \( g'(0) = 2 \), find \( h'(0) \).
2. Differentiate \( f(x) = e^{3g(x)} \) with respect to \( x \).
3. Find the second derivative \( f''(0) \) for the same function \( f(x) = e^{g(x)} \).
4. Solve \( f'(x) \) if \( f(x) = e^{g(x)} + x^2g(x) \).
5. If \( g'(x) = \sin(x) \), find \( f'(x) \) for \( f(x) = e^{g(x)} \).

**Tip**: Always apply the chain rule when differentiating composite functions like \( e^{g(x)} \).

Let f(x) = e^{g(x)}, where g(0) = 6 and g'(0) = 2. Find f'(0).

Learn how to differentiate the function f(x) = e^{g(x)} using the chain rule. Given g(0) = 6 and g'(0) = 2, find the derivative f'(0). This problem covers key concepts in differentiation and exponential functions, suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation