91Ó°ÊÓ

Rational functions are defined as the ratio of two polynomials, but when dealing with trigonometric expressions, things can get a bit tricky. In the given exercise, the expression \( p = \frac{\sin q + \cos q}{\cos q} \) is a rational function where trigonometric functions replace the polynomials. To find the derivative of a rational function, one often can use the quotient rule, but sometimes simplifying the expression first is more convenient. The quotient rule states that for a function \( f(x) = \frac{u(x)}{v(x)} \), the derivative is \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \).
In simpler cases, such as this one, simplifying the expression first can save time and effort, as shown in the step-by-step solution provided. Instead of applying the quotient rule, the function was simplified directly, making it easier to differentiate.

Simplifying trigonometric expressions is a very handy step in calculus, especially when you're about to find derivatives. In the original problem, simplification made the process less cumbersome by transforming a complex rational expression into a simple trigonometric one.
By rewriting \( p = \frac{\sin q + \cos q}{\cos q} \) as \( p = \tan q + 1 \), we can see the function is a lot easier to handle. Here's how:

• Separate the fraction: Divide each term in the numerator by the denominator.
• Simplify: \( \frac{\sin q}{\cos q} = \tan q \) and \( \frac{\cos q}{\cos q} = 1 \), so the expression simplifies to \( \tan q + 1 \).

This simplification turns a daunting task into a straightforward one, leading to a simple differentiation.

Though the chain rule wasn't directly used in the provided solution, it's noteworthy in a broader context when differentiating composed functions. The chain rule is essential when a function is composed with another, such as \( f(g(x)) \), and is expressed as \( \frac{d}{dx}f(g(x)) = f'(g(x))g'(x) \).
Understanding the chain rule helps when you're dealing with more complex trigonometric functions where a direct differentiation isn't feasible. Having a solid understanding of this rule is crucial in calculus, as it helps to break down more complicated derivatives into manageable steps. Although in this situation, the simplification to \( \tan q + 1 \) eliminated the need for the chain rule, knowing it allows you to tackle a wide range of functions with ease in future problems.