Problem 3 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=3 \sin x+5 \cos x $$

Step-by-Step Solution

Answer

The derivative of the function is \( f'(x) = 3\cos x - 5\sin x \).

Step 1: Identify Derivative Rules

To find the derivative of a function of the form \( f(x) = a \sin x + b \cos x \), use the derivative rules: \( \frac{d}{dx}(\sin x) = \cos x \) and \( \frac{d}{dx}(\cos x) = -\sin x \).

Step 2: Differentiate Each Term Separately

Differentiate each term of the function separately. The derivative of \( 3\sin x \) is \( 3 \cos x \) because the derivative of \( \sin x \) is \( \cos x \). The derivative of \( 5\cos x \) is \( -5 \sin x \) because the derivative of \( \cos x \) is \( -\sin x \).

Step 3: Combine the Derivatives

Combine the derivatives of each term to get the derivative of the entire function. Thus, the derivative \( f'(x) = 3\cos x - 5\sin x \).

Key Concepts

Trigonometric Functions

Trigonometric functions play a pivotal role in calculus, especially when finding derivatives. The two most common trigonometric functions are sine (\( \sin x \)) and cosine (\( \cos x \)). These functions are periodic and describe the ratio of different sides of a right triangle relative to one of its angles.

• The sine function reflects how the length of the opposite side varies with the angle in a right triangle.
• The cosine function represents how the length of the adjacent side changes.

In calculus, derivatives of these functions are particularly important. The derivative of \( \sin x \) is \( \cos x \), which means it measures how the sine function's slope changes at every point. Conversely, the derivative of \( \cos x \) is \(-\sin x \). This reflects the decreasing nature of the cosine function as its angle increases. Understanding these derivatives is crucial when solving calculus problems involving trigonometry.

Chain Rule

The chain rule is a fundamental principle in calculus for finding derivatives, especially for functions within other functions, often referred to as composite functions. This rule can be thought of as a "derivative of a derivative," allowing us to handle complex expressions more easily.

• Whenever a function is composed of multiple layers, like \( \sin(5x) \) or \( \cos(x^2) \), the chain rule helps us differentiate systematically.
• The rule states that to differentiate a composite function like \( f(g(x)) \), you multiply the derivative of the outer function by the derivative of the inner function. Mathematically, \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \).

Though not directly used in our current example, being adept with the chain rule is crucial for handling more intricate trigonometric derivatives where arguments themselves are functions of a variable.

Calculus

Calculus is the branch of mathematics that deals with change and motion. It is divided into two main areas: differentiation and integration. In our problem, we focus on differentiation, which involves finding the derivative of a function. A derivative represents how a function changes as its input changes. It's like looking at the slope of a line that best describes the function at any given point.

• Differentiation involves several rules and techniques, such as the power rule, product rule, and our focus, the trigonometric rules.
• Understanding how functions change through differentiation helps solve real-world problems, like calculating speed or predicting future trends based on current data.

By mastering calculus, especially differentiation, students can gain powerful insights into the behavior of various functions, providing them a toolkit for tackling complex mathematical and practical problems.