Let's start by rewriting the function for clarity:

\[
f(x) = \frac{1}{\cos\left(\frac{x}{2}\right)}
\]

Now, we need to differentiate this function with respect to \( x \). To do so, we will apply the chain rule and the derivative of a trigonometric function. Here's the process:

### Step 1: Rewrite as a power of cosine
\[
f(x) = \left(\cos\left(\frac{x}{2}\right)\right)^{-1}
\]

### Step 2: Apply the chain rule
The chain rule tells us to first differentiate the outer function and then the inner function. The outer function is \(\cos\left(\frac{x}{2}\right)^{-1}\), and its derivative is:

\[
\frac{d}{dx} \left( u^{-1} \right) = -u^{-2} \cdot \frac{du}{dx}
\]

So,

\[
\frac{d}{dx} \left( \cos\left(\frac{x}{2}\right)^{-1} \right) = -\left(\cos\left(\frac{x}{2}\right)\right)^{-2} \cdot \frac{d}{dx} \left( \cos\left(\frac{x}{2}\right) \right)
\]

### Step 3: Differentiate \(\cos\left(\frac{x}{2}\right)\)
The derivative of \(\cos\left(\frac{x}{2}\right)\) is:

\[
\frac{d}{dx} \left( \cos\left(\frac{x}{2}\right) \right) = -\sin\left(\frac{x}{2}\right) \cdot \frac{1}{2}
\]

### Step 4: Combine results
Substitute the derivative of \(\cos\left(\frac{x}{2}\right)\) back into the previous expression:

\[
f'(x) = -\left(\cos\left(\frac{x}{2}\right)\right)^{-2} \cdot \left( -\sin\left(\frac{x}{2}\right) \cdot \frac{1}{2} \right)
\]

Simplifying:

\[
f'(x) = \frac{\sin\left(\frac{x}{2}\right)}{2 \cos^2\left(\frac{x}{2}\right)}
\]

Thus, the derivative of \( f(x) = \frac{1}{\cos\left(\frac{x}{2}\right)} \) is:

\[
f'(x) = \frac{\sin\left(\frac{x}{2}\right)}{2 \cos^2\left(\frac{x}{2}\right)}
\]

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

Here are 5 related questions to deepen understanding:

1. How would you differentiate a function like \( \frac{1}{\sin\left(\frac{x}{3}\right)} \)?
2. What happens if we differentiate \( f(x) = \tan\left(\frac{x}{2}\right) \)?
3. How can the derivative of \( \sec(x) \) be derived using a similar method?
4. How does the chain rule apply when differentiating trigonometric functions?
5. What is the significance of second derivatives in this context?

**Tip:** Always simplify trigonometric functions before differentiating—it makes applying the chain rule easier.

This page provides a detailed step-by-step solution to finding the derivative of f(x) = 1/cos(x/2). The solution involves using the chain rule and differentiating trigonometric functions, making it suitable for students in Grades 11-12 studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation