Problem 9 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=2 \sin (3 x+1) $$

Step-by-Step Solution

Answer

The derivative is \( 6\cos(3x + 1) \).

Step 1: Identify the Function

We have the function \( f(x) = 2\sin(3x + 1) \). This is a composition of functions where the outer function is \( 2\sin(u) \) with \( u = 3x + 1 \).

Step 2: Differentiate the Outer Function

Differentiate the outer function \( 2\sin(u) \) with respect to \( u \). The derivative of \( \sin(u) \) is \( \cos(u) \), so the derivative of \( 2\sin(u) \) is \( 2\cos(u) \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( u = 3x + 1 \) with respect to \( x \). The derivative is \( \frac{du}{dx} = 3 \).

Step 4: Apply the Chain Rule

Use the chain rule to find the derivative of the composite function. Combine the derivatives from Steps 2 and 3: \( \frac{df}{dx} = \frac{d}{du}(2\sin(u)) \cdot \frac{du}{dx} = 2\cos(u) \cdot 3 \).

Step 5: Substitute Back the Inner Function

Substitute back \( u = 3x + 1 \) into the derivative we found. Therefore, \( \frac{df}{dx} = 2 \cdot 3 \cdot \cos(3x + 1) = 6 \cos(3x + 1) \).

Key Concepts

Understanding Derivatives

A derivative is a fundamental concept in calculus that reveals the rate at which a function changes. In simpler terms, the derivative tells us how a function behaves as its input values change. For beginners, you can think of it like the slope of a line on a graph. The steeper the line, the faster the function value is changing.
For functions of one variable, the process of finding the derivative is often called differentiation. Differentiating a function involves using certain rules and techniques tailored to the types of functions you encounter.
Some basic rules include the power rule, product rule, quotient rule, and the chain rule, which we'll explore more deeply later. When dealing with trigonometric functions like sine or cosine, remember their specific derivatives:

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

By practicing various problems, you'll get a better intuition for how these rules work and how to apply them effectively in different scenarios.

The Power of the Chain Rule

The chain rule is a crucial tool in calculus used for differentiating composite functions. A composite function is when one function is nested inside another, like our function \( f(x) = 2\sin(3x + 1) \). To differentiate such a function, we need to "chain" the derivatives together.
The chain rule states that how quickly the composite function changes is determined by the product of how quickly the outer function changes and how quickly the inner function changes. In other words:

• If a function \( y = f(g(x)) \), then its derivative \( \frac{dy}{dx} \) is given by \( \frac{df}{dg} \cdot \frac{dg}{dx} \)

Let's break it down using our example:

• First, identify the inner function and outer function. Here, \( u = 3x + 1 \) is the inner function and the sine function is the outer function.
• Differentiating the outer function, \( 2\sin(u) \), with respect to \( u \) gives us \( 2\cos(u) \).
• Differentiating the inner function, \( 3x + 1 \), with respect to \( x \) gives us \( 3 \).
• Finally, applying the chain rule means multiplying these derivatives together: \( 2\cos(u) \cdot 3 \).

This multiplication gives us the overall rate of change for our original function, showing how vital the chain rule is for handling nested functions.

Trigonometric Functions in Calculus

Trigonometric functions such as \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \) often appear in calculus problems due to their periodic nature and wide applicability in modeling waves, oscillations, and circular motion. Understanding their derivatives is fundamental in calculus.
Here are the basic derivatives for trigonometric functions:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• The derivative of \( \tan(x) \) is \( \sec^2(x) \).

These derivatives form the building blocks for more complex differentiation problems, including those involving the chain rule, product rule, or quotient rule.
When working with trigonometric functions:

• Identify any composite structures, such as transformations like \( 3x + 1 \) within sine or cosine.
• Remember their cyclic properties, which can help in understanding their graphs and periodic nature.

By mastering these derivatives and the behaviors of trigonometric functions, you'll be better equipped to handle a wide array of problems in calculus involving cycles and oscillations.