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To derive the given trigonometric derivative rule \( \frac{d}{dx} \sec(x) = \sec(x) \tan(x) \) using the quotient rule, let's proceed step by step.

### Step 1: Express \(\sec(x)\) in terms of \(\cos(x)\)
We know that:
\[
\sec(x) = \frac{1}{\cos(x)}
\]
We will apply the quotient rule to differentiate this expression.

### Step 2: Quotient Rule Formula
The quotient rule for differentiation states that for two functions \( f(x) \) and \( g(x) \),
\[
\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}
\]
Here, we have \( f(x) = 1 \) and \( g(x) = \cos(x) \).

### Step 3: Differentiate Using the Quotient Rule
Now, applying the quotient rule:

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

Simplifying this expression:

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

### Step 4: Simplify the Result
We know that:
\[
\frac{\sin(x)}{\cos(x)} = \tan(x)
\quad \text{and} \quad \frac{1}{\cos(x)} = \sec(x)
\]
Thus, we can rewrite the expression as:
\[
\frac{d}{dx} \sec(x) = \sec(x) \tan(x)
\]

This completes the derivation of the rule:
\[
\frac{d}{dx} \sec(x) = \sec(x) \tan(x)
\]

Would you like any further clarification or additional details? Here are some related questions you might explore:

1. How do you apply the quotient rule to more complex functions?
2. What is the geometric interpretation of the secant function's derivative?
3. Can we derive the derivative of \(\sec(x)\) using other methods like implicit differentiation?
4. How can we extend this result to higher-order derivatives of \(\sec(x)\)?
5. What are some real-world applications where the derivative of \(\sec(x)\) is useful?

**Tip:** Always rewrite trigonometric functions in terms of sine and cosine for easier differentiation!

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

Learn how to derive the derivative of sec(x) using the quotient rule in calculus. This step-by-step solution walks through applying the quotient rule to the secant function, breaking down key trigonometric identities such as sec(x) and tan(x). Ideal for students in advanced high school or introductory college-level calculus courses.

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