Problem 57 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{\sec x^{2}}{\sec ^{2} x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -\sin x \).

Step 1: Understand the Function

The function given is \( f(x) = \frac{\sec x^2}{\sec^2 x} \), which can be simplified before finding its derivative. Simplifying the expression will make the differentiation easier.

Step 2: Simplify the Function

Simplify the function: \[ f(x) = \frac{\sec x^2}{\sec^2 x} = \frac{1}{\sec x} = \cos x \] since \( \sec x = \frac{1}{\cos x} \). Our function is now \( f(x) = \cos x \).

Step 3: Differentiate the Function

Differentiate \( f(x) = \cos x \) with respect to \( x \). The derivative \( \frac{d}{dx}(\cos x) \) is \( -\sin x \). Therefore, \( f'(x) = -\sin x \).

Step 4: Final Step: State the Derivative

The derivative of the function \( f(x) = \cos x \) is \( f'(x) = -\sin x \).

Key Concepts

Simplification

Simplification in calculus is all about making things easier. It helps to turn complex equations into simpler forms for easy manipulation. Simplification plays a crucial role when you have a function like \( f(x) = \frac{\sec x^2}{\sec^2 x} \). Before jumping into differentiation, it's wise to simplify the function.How do you simplify it? Well, start by knowing your trigonometric identities. The secant function, \( \sec x \), is the reciprocal of cosine: \( \sec x = \frac{1}{\cos x} \). Thus, when you have \( \frac{\sec x^2}{\sec^2 x}, \) you can simplify it.

• First, express secant in terms of cosine, meaning \( \sec x = 1/\cos x \).
• Then observe the function: \( \frac{\sec x^2}{\sec^2 x} = \frac{1}{\sec x} \).

Finally, using the secant-cosine identity, \( \frac{1}{\sec x} = \cos x \). Now, the function looks much cleaner and is ready for differentiation.

Trigonometric Functions

Trigonometric functions form the backbone of many calculus problems. Functions like sine, cosine, and secant are key players in calculus, especially when dealing with periodic phenomena.When you see a function like \( f(x) = \frac{\sec x^2}{\sec^2 x} \), you need to know what secant means. As mentioned:

• Secant (\( \sec x \)) is the reciprocal of cosine.
• \( \sec x = \frac{1}{\cos x} \)
• Therefore, \( \frac{1}{\sec x} = \cos x \).

Understanding these relationships helps in recognizing and transforming trigonometric functions into easier forms. Also, once simplified to \( \cos x \), you immediately recognize that it's one of the basic trigonometric functions, making further steps like differentiation much simpler.

Differentiation Rules

Differentiation rules tell us how to find the rate of change of functions, which is central to calculus. With our simplified function, \( f(x) = \cos x \), differentiation becomes straightforward.There are specific rules for differentiating trigonometric functions. Let’s focus on cosine:

• The derivative of \( \cos x \) with respect to \( x \) is \( -\sin x \).
• This is a standard result from basic differentiation rules, helping us understand changes in trigonometric curves.

Knowing these rules is critical for solving calculus problems. Without understand them, calculating derivatives correctly would be tough.In our example, after simplification to \( \cos x \), applying the differentiation rule \( \frac{d}{dx}(\cos x) = -\sin x \) gives the derivative. Thus, the derivative of \( f(x) = \cos x \) is \( f'(x) = -\sin x \). These rules streamline the process and allow for effective problem-solving in calculus.