To find the derivative of \( f(x) = \cos(g(x)) \), we’ll use the chain rule.

### Solution:
Given:
\[
f(x) = \cos(g(x))
\]

1. **Differentiate the outer function** \( \cos(g(x)) \) with respect to \( g(x) \):
   \[
   \frac{d}{dg} \cos(g(x)) = -\sin(g(x))
   \]

2. **Multiply by the derivative of the inner function** \( g(x) \) with respect to \( x \):
   \[
   f'(x) = -\sin(g(x)) \cdot g'(x)
   \]

### Final Answer:
\[
f'(x) = -\sin(g(x)) \cdot g'(x)
\]

Would you like further clarification on any step?

---

Here are some related questions to deepen your understanding:
1. How would the derivative change if \( f(x) = \sin(g(x)) \)?
2. What if \( f(x) = \cos(x^2 + 3x) \)? How would you find the derivative?
3. How does the chain rule apply when differentiating \( f(x) = \cos(g(h(x))) \)?
4. What is the second derivative of \( f(x) = \cos(g(x)) \)?
5. How would you handle the derivative if \( g(x) \) were a trigonometric function like \( \sin(x) \) or \( \tan(x) \)?

**Tip:** The chain rule is essential for differentiating compositions of functions, especially in trigonometric, exponential, and logarithmic contexts.

Learn how to find the derivative of f(x) = cos(g(x)) by applying the chain rule. This solution includes a step-by-step process for differentiating trigonometric functions and compositions of functions, suitable for high school 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