Problem 3 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=3 \sin x+5 \cos x-2 \sec x $$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = 3 \cos x - 5 \sin x - 2 \sec x \tan x\).

Step 1: Identify Components of the Function

The function to differentiate is given as \(f(x) = 3 \sin x + 5 \cos x - 2 \sec x\). Note that it consists of three components: \(3 \sin x\), \(5 \cos x\), and \(-2 \sec x\). Each of these components needs to be differentiated separately.

Step 2: Differentiate Each Component

Using the derivative rules: - The derivative of \(a \sin x\) is \(a \cos x\), so \(\frac{d}{dx}[3 \sin x] = 3 \cos x\).- The derivative of \(a \cos x\) is \(-a \sin x\), so \(\frac{d}{dx}[5 \cos x] = -5 \sin x\).- The derivative of \(a \sec x\) is \(a \sec x \tan x\), so \(\frac{d}{dx}[-2 \sec x] = -2 \sec x \tan x\).

Step 3: Combine the Derivatives

Add the derivatives of each component to obtain the derivative of the whole function:\[ f'(x) = 3 \cos x - 5 \sin x - 2 \sec x \tan x \].

Key Concepts

Trigonometric Functions

Understanding trigonometric functions is crucial, as they form essential building blocks for mathematics, especially in calculus. In the exercise, we encounter sine, cosine, and secant functions. These functions help relate angles in right triangles to ratios of side lengths.

• Sine (\( \sin x \) denotes the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\( \cos x \)) is the ratio of the adjacent side to the hypotenuse.
• Secant (\( \sec x \)) is the reciprocal of cosine, given by \( \sec x = \frac{1}{\cos x} \).

These functions are periodic, meaning they repeat their values in regular intervals, which is why they're represented in wave-like graphs. Mastering their properties is key to solving many calculus problems.
Understanding these trigonometric functions allows us to solve for derivatives, which tell us how these functions change over time.

Differentiation Rules

Differentiation rules guide us in finding the derivatives of various functions efficiently. They provide formulas that make the process of finding derivatives systematic. In the given problem, we apply differentiation rules to trigonometric functions:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \(-\sin x \).
• The derivative of \( \sec x \) is \( \sec x \tan x \).

Since the function is composed of a combination of trigonometric functions multiplied by constants, the following rules also apply:

• Constant Multiple Rule: The derivative of \( a f(x) \) is \( a f'(x) \), where \( a \) is a constant.
• Sum Rule: The derivative of a sum \( f(x) + g(x) \) is \( f'(x) + g'(x) \).

Applying these rules makes it easier to handle complex expressions like the one in the exercise. This systematic approach ensures we accurately compute the rate of change for each part of the function.

Step-by-Step Solution

Tackling derivative problems with a step-by-step approach ensures clarity and accuracy. Let's walk through the exercise solution.

**Step 1: Break Down the Function**
The function \( f(x) = 3 \sin x + 5 \cos x - 2 \sec x \) is separated into individual components: \( 3 \sin x \), \( 5 \cos x \), and \( -2 \sec x \). Each component involves a single trigonometric function.

**Step 2: Apply Differentiation Rules**
Using known rules:

• The derivative of \( 3 \sin x \) becomes \( 3 \cos x \).
• For \( 5 \cos x \), it transforms to \( -5 \sin x \).
• The term \( -2 \sec x \) differentiates to \( -2 \sec x \tan x \).

**Step 3: Combine Derivatives**
Together, the derivatives form the expression \( f'(x) = 3 \cos x - 5 \sin x - 2 \sec x \tan x \).

This step-by-step breakdown simplifies complex calculus problems, guiding students in using differentiation rules correctly and effectively.