https://file.aiquickdraw.com/mathbot/file/1729127672435icc0c7ag.jpg

To differentiate \( f(x) = 9e^{x \cos x} \), we will apply the chain rule.

### Steps:

1. **Let \( u(x) = x \cos x \)**. First, differentiate \( u(x) \) using the product rule:
   \[
   u'(x) = \frac{d}{dx} \left( x \cos x \right) = \cos x - x \sin x
   \]
   
2. **Differentiate \( f(x) \)**. Now, apply the chain rule to differentiate the exponential function:
   \[
   f'(x) = 9 \cdot e^{x \cos x} \cdot \frac{d}{dx} \left( x \cos x \right)
   \]
   Using the result from step 1:
   \[
   f'(x) = 9e^{x \cos x} \cdot \left( \cos x - x \sin x \right)
   \]

So, the derivative is:
\[
f'(x) = 9e^{x \cos x} \left( \cos x - x \sin x \right)
\]

Would you like a detailed breakdown of the differentiation process or further explanation?

### Related Questions:
1. What is the product rule and when is it used?
2. How do you apply the chain rule in differentiation?
3. How would you differentiate other functions like \( x e^{x \sin x} \)?
4. What is the derivative of trigonometric functions like \( \cos x \) and \( \sin x \)?
5. How does the derivative of exponential functions change with different exponents?

### Tip:
When using the chain rule, always identify the inner and outer functions first, which will guide you through the differentiation steps.

Differentiate the function f(x) = 9e^{x \cos x}.

Learn how to differentiate the function f(x) = 9e^{x cos x} using the chain rule and product rule. This solution provides a step-by-step explanation for applying these rules in calculus, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation