We are tasked with finding the derivative of the following expression:

$f(x) = \frac{\sec(x)}{1 + \tan(x)}$

Steps:

1. Apply the Quotient Rule:
   The quotient rule states that if $f(x) = \frac{g(x)}{h(x)}$, then the derivative is given by: $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$ Here, $g(x) = \sec(x)$ and $h(x) = 1 + \tan(x)$.

2. Find $g'(x)$:
   The derivative of $g(x) = \sec(x)$ is: $g'(x) = \sec(x)\tan(x)$

3. Find $h'(x)$:
   The derivative of $h(x) = 1 + \tan(x)$ is: $h'(x) = \sec^2(x)$

4. Apply the quotient rule:
   Now substitute into the quotient rule formula: $f'(x) = \frac{\sec(x) \sec^2(x) - \sec(x)\tan(x) (1 + \tan(x))}{(1 + \tan(x))^2}$

5. Simplify: $f'(x) = \frac{\sec(x) [\sec^2(x) - \tan(x)(1 + \tan(x))]}{(1 + \tan(x))^2}$ Expanding the terms inside the bracket: $f'(x) = \frac{\sec(x) [\sec^2(x) - \tan(x) - \tan^2(x)]}{(1 + \tan(x))^2}$

This is the simplified form of the derivative.

Would you like further simplification or details on any step?

Related Questions:

1. What is the derivative of $\sec(x)$?
2. Can the quotient rule be applied to any function division?
3. What is the significance of the secant and tangent functions in trigonometry?
4. How do we simplify trigonometric expressions?
5. Can the quotient rule be used with product rule for more complex functions?

Tip: When using the quotient rule, always carefully differentiate both the numerator and denominator separately before applying the formula.