Problem 2 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative is \(-3\sin x - 2\cos x\).

Step 1: Understanding the Problem

We need to find the derivative of the function \( f(x) = 3 \cos x - 2 \sin x \) with respect to \( x \). This involves using the rules of differentiation for trigonometric functions.

Step 2: Differentiation Rule for Cosine Function

The derivative of \( \cos x \) with respect to \( x \) is \( -\sin x \). Since the term \( 3\cos x \) involves a constant multiplier, apply the constant rule which tells us to keep the constant as is and take the derivative of \( \cos x \). So, the derivative of \( 3\cos x \) is \( 3(-\sin x) = -3\sin x \).

Step 3: Differentiation Rule for Sine Function

The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \). Similarly, for the term \(-2\sin x\), apply the constant rule. The derivative of \(-2\sin x\) becomes \(-2\cos x\).

Step 4: Combining the Derivatives

After finding the derivatives of each term, combine them to get the final result. So, the derivative \( f'(x) \) is \( -3\sin x - 2\cos x \).

Step 5: Final Result

The derivative of the function \( f(x) = 3\cos x - 2\sin x \) is \( f'(x) = -3\sin x - 2\cos x \). This completes the differentiation process.

Key Concepts

Trigonometric Functions and Their Properties

Trigonometric functions like sine and cosine are fundamental in calculus and many scientific fields. They represent relationships in right-angled triangles and are periodic in nature.
The cosine function, \( \cos x \), specifically relates the adjacent side to the hypotenuse in a right triangle. It oscillates between -1 and 1. The sine function, \( \sin x \), relates the opposite side to the hypotenuse and similarly oscillates between -1 and 1.

• Both functions have a period of \( 2\pi \), meaning they repeat every \( 2\pi \) radians.
• Important identities include \( \sin^2 x + \cos^2 x = 1 \), which are useful in simplifying expressions.

Understanding these functions is critical when applying differentiation as it provides insight into how they change over angles.

Differentiation Rules for Trigonometric Functions

Differentiation is the process of finding the rate at which a function is changing at any given point. For trigonometric functions, specific rules are followed.
The derivative of \( \cos x \) with respect to \( x \) is \( -\sin x \). This tells us how the rate of change of cosine relates to sine. Similarly, the derivative of \( \sin x \) with respect to \( x \) is \( \cos x \).

• When a function is multiplied by a constant, like \( 3\cos x \), the constant remains and the function is differentiated as usual. Thus, \( 3\cos x \) becomes \( -3\sin x \).
• The same rule applies to \(-2\sin x\), which becomes \(-2\cos x\) after differentiation.

This knowledge helps in finding derivatives of functions involving trig terms, just like in the original problem. By applying these rules, the function \( f(x) = 3\cos x - 2\sin x \) is differentiated step-by-step.

Calculus and Derivative Applications

Calculus, as a broader field, explores how things change and lets us understand and describe these changes mathematically.
Differentiation, in particular, answers questions like how fast something is changing at a certain point. It involves concepts like limits to find derivatives of functions. The process used in the solution of the exercise is part of this larger framework.

• Finding the derivative, like we did for \( f(x) = 3\cos x - 2\sin x \), shows us the slope of the tangent line at any point \( x \).
• This has practical applications in physics for determining velocity or acceleration and in economics to calculate marginal rates of change.

Using calculus, students can solve real-world problems by anticipating how they will evolve. It begins by mastering the core ideas of differentiation and its rules, especially for functions like trigonometric ones.