Deriving the Derivative of sec(x) Using the Quotient Rule

Math Problem Statement

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

Solution

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:

\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!

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Trigonometry

Formulas

Quotient rule: (d/dx)(f(x)/g(x)) = (g(x)f'(x) - f(x)g'(x)) / [g(x)]^2
sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x)

Theorems

Quotient Rule
Derivative of Secant Function

Suitable Grade Level

Grades 11-12, College-level Calculus

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