Problem 7 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative is \( 3\cos(3x) \).

Step 1: Recall the Derivative Rule for Sine Function

The derivative of the sine function with respect to its argument is given by \( \frac{d}{du}(\sin u) = \cos u \), where \( u \) is a function of \( x \).

Step 2: Identify the Inner Function

In this problem, the function inside the sine is \( u = 3x \). Thus, \( \frac{d}{dx}(\sin(3x)) = \cos(3x) \cdot \frac{du}{dx} \).

Step 3: Differentiate the Inner Function

The derivative of \( u = 3x \) with respect to \( x \) is \( \frac{du}{dx} = 3 \).

Step 4: Apply the Chain Rule

Substitute the derivatives into the chain rule formula: \( \frac{d}{dx}(\sin(3x)) = \cos(3x) \times 3 \).

Step 5: Simplify the Expression

Combine the terms from the chain rule: \( \frac{d}{dx}(\sin(3x)) = 3\cos(3x) \).

Key Concepts

Sine Function

The sine function, denoted as \( \sin(x) \), is one of the fundamental trigonometric functions used in mathematics to describe oscillations and waves. It's a periodic function, meaning it repeats its values in regular intervals or periods. The sine function has a range between -1 and 1, and its graph is a smooth wave-like pattern called a sine wave.

• Basis of circular motion and hence vital for trigonometry.
• Used to model periodic phenomena such as sound and light waves.

The mathematical definition relies on the unit circle: the \( y \)-coordinate of a point on the unit circle corresponding to an angle measure of \( x \) gives \( \sin(x) \). Understanding the derivative of the sine function is essential, especially in calculus, as it helps predict the rate of change of rhythms modeled by the sine curve.

Chain Rule

The chain rule is a critical theorem in calculus used to differentiate composites of functions. To put it simply, when you have a function inside another function, the chain rule allows you to find the derivative of this composite function. It is tremendously helpful when dealing with a function like \( \sin(3x) \), where you have sine (a trigonometric function) applied to \( 3x \) (a linear function).The chain rule can be formulated as:
If you have a composite function \( g(f(x)) \), then the derivative \( \frac{d}{dx}[g(f(x))] = g'(f(x)) \cdot f'(x) \). For our problem:

• \( u = 3x \) is the inner function \( f(x) \).
• \( \sin(u) \) is the outer function \( g(u) \).

By applying the chain rule, you differentiate the outer function \( \sin(u) \) to get \( \cos(u) \), then multiply it by the derivative of the inner function, which is \( \frac{du}{dx} = 3 \). Hence, you get \( 3\cos(3x) \).

Calculus

Calculus is a branch of mathematics that studies continuous change. It is divided into two main branches: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which represents the rate of change of a quantity.
Derivatives are used extensively in various fields, such as physics, engineering, economics, and beyond, whenever there's a need to understand how things change. The approach to finding derivatives involves rules and techniques, like the chain rule, the product rule, and the quotient rule.

• Derivatives: Provide information about the shape of a function's graph. They help determine maxima, minima, and points of inflection.
• Fundamental for understanding motion and change: Used in physics to understand velocities and accelerations.

In the context of this exercise, calculus enables us to understand how the sine function \( \sin(3x) \) changes as \( x \) varies, allowing us to calculate its derivative as \( 3\cos(3x) \). This insight is invaluable for modeling and solving real-world problems.