Use the quotient rule to derive the following trigonometric derivative rule: (d/dx) (sec(x)) = sec(x) tan(x)

To derive the derivative of \( \sec(x) \) using the **quotient rule**, we start by expressing \( \sec(x) \) in terms of sine and cosine. Recall that:

\[
\sec(x) = \frac{1}{\cos(x)}
\]

Now, we will apply the **quotient rule**, which states that if \( f(x) = \frac{g(x)}{h(x)} \), then:

\[
\frac{d}{dx} \left( \frac{g(x)}{h(x)} \right) = \frac{h(x)g'(x) - g(x)h'(x)}{(h(x))^2}
\]

### Step 1: Set up the quotient rule
For \( \sec(x) = \frac{1}{\cos(x)} \), we have:
- \( g(x) = 1 \)
- \( h(x) = \cos(x) \)

Now, let's differentiate:

- \( g'(x) = 0 \) because the derivative of a constant (1) is 0.
- \( h'(x) = \frac{d}{dx}[\cos(x)] = -\sin(x) \).

### Step 2: Apply the quotient rule
Substitute these into the quotient rule formula:

\[
\frac{d}{dx} \left( \frac{1}{\cos(x)} \right) = \frac{\cos(x)(0) - 1(-\sin(x))}{(\cos(x))^2}
\]

This simplifies to:

\[
\frac{d}{dx} \left( \sec(x) \right) = \frac{\sin(x)}{\cos^2(x)}
\]

### Step 3: Express in terms of secant and tangent
Next, recall that:

\[
\tan(x) = \frac{\sin(x)}{\cos(x)} \quad \text{and} \quad \sec(x) = \frac{1}{\cos(x)}
\]

Thus, we can rewrite \( \frac{\sin(x)}{\cos^2(x)} \) as:

\[
\frac{\sin(x)}{\cos^2(x)} = \frac{1}{\cos(x)} \cdot \frac{\sin(x)}{\cos(x)} = \sec(x) \tan(x)
\]

### Final result:
\[
\frac{d}{dx} \left( \sec(x) \right) = \sec(x) \tan(x)
\]

Thus, the derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \), as required.

---

Would you like more details on any of the steps? Here are five follow-up questions:

1. How does the quotient rule differ from the product rule in differentiation?
2. Can you explain how the derivative of \( \tan(x) \) is found using the quotient rule?
3. What is the geometrical interpretation of the derivative of \( \sec(x) \)?
4. How can we derive the derivative of \( \csc(x) \) using a similar approach?
5. What are some real-world applications of the secant function in trigonometry?

**Tip**: Memorizing the derivatives of basic trigonometric functions will help you solve complex problems faster!

Explore the step-by-step process of deriving the derivative of sec(x) using the quotient rule in calculus. This problem is perfect for students in Grades 11-12 who are learning about trigonometric functions and differentiation.

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