Problem 15 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative of \( f(x) = 3 \sin(x^2) \) is \( 6x \cos(x^2) \).

Step 1: Understanding the Function

The function given is \( f(x) = 3 \sin(x^2) \). We need to find the derivative with respect to \( x \). The function involves a trigonometric function \( \sin \) and a composition inside \( \sin \) with \( x^2 \).

Step 2: Applying the Chain Rule

To differentiate \( f(x) = 3 \sin(x^2) \), we use the chain rule. The chain rule is: \(( g(f(x)) )' = g'(f(x)) \cdot f'(x) \). Here, let \( g(u) = 3 \sin(u) \) with \( u = x^2 \). Thus, \( g'(u) = 3 \cos(u) \).

Step 3: Differentiating the Inner Function

Next, differentiate the inner function \( u = x^2 \) with respect to \( x \). The derivative is \( \frac{du}{dx} = 2x \).

Step 4: Combining Results from Chain Rule

Apply the chain rule: \[ \frac{df}{dx} = g'(u) \cdot \frac{du}{dx} = 3 \cos(x^2) \cdot 2x = 6x \cos(x^2). \]

Step 5: Writing the Final Derivative

The derivative of \( f(x) = 3 \sin(x^2) \) with respect to \( x \) is \( \frac{df}{dx} = 6x \cos(x^2) \). This is our final answer.

Key Concepts

Derivatives

Derivatives play a crucial role in calculus, helping us understand how functions change. A derivative tells us the rate of change of a function with respect to an independent variable, usually denoted as \( x \). In our problem, we are looking at how the function \( f(x) = 3 \sin(x^2) \) changes as \( x \) changes.
In a broader sense, when you take a derivative of a function, you are essentially finding its slope at any given point. This is useful for determining trends, directions, and rates of increase or decrease.

• The derivative of a constant is zero.
• The derivative of \( x^n \), where \( n \) is any real number, is \( nx^{n-1} \).
• Understanding derivatives opens the door to analyzing real-world phenomena.

The task at hand is made more interesting because it involves not just any function, but a trigonometric function composed with another function, making it a perfect candidate for using advanced techniques like the Chain Rule.

Chain Rule

The Chain Rule is a fundamental technique in calculus used for differentiating complex functions, especially when one function is nested inside another. It's particularly useful for functions like \( 3 \sin(x^2) \) where you have a function within a function. The Chain Rule can be stated as: if you have \( y = g(f(x)) \), then the derivative \( \frac{dy}{dx} = g'(f(x)) \cdot f'(x) \).
This is exactly what we use to differentiate \( 3 \sin(x^2) \). Here’s a quick recap of the steps:

• Identify the outer function and the inner function.
• In our problem, the outer function is \( 3 \sin(u) \), and the inner function is \( u = x^2 \).
• Differentiating \( 3 \sin(u) \) gives us \( 3 \cos(u) \).
• Differentiating \( u = x^2 \) yields \( 2x \).

By applying the Chain Rule, the final derivative of \( 3 \sin(x^2) \) becomes \( 6x \cos(x^2) \).
Understanding the Chain Rule empowers you to tackle more complicated derivative problems with ease.

Trigonometric Functions

Trigonometric functions are essential components in calculus and geometry, revolving around the angles and sides of triangles. Common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). In our problem, we deal with the sine function.
These functions are periodic, meaning they repeat their values in a regular pattern.

• For example, \( \sin(x) \) oscillates between -1 and 1.
• The derivative of \( \sin(x) \) is \( \cos(x) \), which is also a trigonometric function.
• Trigonometric functions are invaluable in solving real-life problems involving waves and cycles.

When finding the derivative of functions involving trigonometric parts, it's vital to remember these derivatives and how they interact with other rules in calculus.
The exercise focused on the function \( 3 \sin(x^2) \), illustrating how a trigonometric function can be differentiated using both its inherent properties and the Chain Rule for the most accurate results.