Problem 8 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=e^{2 x / 3}\end{equation}

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = \frac{2}{3} e^{\frac{2x}{3}} \).

Step 1: Identify the Function

The function given is \[ y = e^{\frac{2x}{3}} \]This is an exponential function where the exponent is a linear function of \(x\). We need to find its derivative with respect to \(x\).

Step 2: Apply the Chain Rule

To differentiate \( y = e^{\frac{2x}{3}} \), use the chain rule. The chain rule states that\[ \frac{dy}{dx} = \frac{d}{dx} \left( e^{u} \right) \cdot \frac{du}{dx}, \]where \( u = \frac{2x}{3} \).

Step 3: Differentiate the Exponential Function

The derivative of \( e^{u} \) with respect to \( u \) is itself, \( e^{u} \). Therefore, \[ \frac{d}{du} \left( e^{\frac{2x}{3}} \right) = e^{\frac{2x}{3}}. \]

Step 4: Differentiate the Inner Function

Now, find the derivative of \( u = \frac{2x}{3} \) with respect to \( x \). The derivative is a constant multiplied by a variable, so \[ \frac{du}{dx} = \frac{2}{3}. \]

Step 5: Combine the Results

Combining the results from step 3 and step 4 using the chain rule gives:\[ \frac{dy}{dx} = e^{\frac{2x}{3}} \cdot \frac{2}{3}. \] Thus, \[ \frac{dy}{dx} = \frac{2}{3} e^{\frac{2x}{3}}. \]

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In our exercise, the function is given by \( y = e^{\frac{2x}{3}} \). Here, \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
In exponential functions:

• The base \( e \) remains constant.
• The variable (in this case \( x \)) is in the exponent position.
• Exponential functions are significant in many real-world applications such as compound interest, population growth, and more.

They have unique properties like:

• The rate of change (or growth rate) is proportional to the value of the function itself.
• The derivative of \( e^u \) with respect to \( u \) is itself, \( e^u \).

This special property makes exponential functions easier to differentiate and apply the chain rule.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions, which are functions composed of other functions. Suppose you have a function \( y \) that includes another function \( u(x) \) inside it. The chain rule helps find the derivative of such functions.
A typical scenario is when you have \( y = f(g(x)) \). The chain rule states:

• Differentiate the outer function \( f \) with respect to the inner function \( u \).
• Multiply this derivative by the derivative of the inner function \( u \) with respect to the variable, \( x \).

For our exercise, where \( y = e^{\frac{2x}{3}} \), the chain rule is applied:

• \( u = \frac{2x}{3} \), an inner linear function.
• The derivative of \( u \) is \( \frac{2}{3} \).

Combine these using the chain rule to get the overall derivative \( \frac{dy}{dx} = e^{\frac{2x}{3}} \cdot \frac{2}{3} \). This demonstrates how the chain rule simplifies the differentiation of composite functions.

Calculus

Calculus is the branch of mathematics that explores concepts like change and motion, primarily through the study of derivatives and integrals. It provides a framework for modeling systems with changing quantities.
Key areas in calculus include:

• Differentiation: Focuses on how functions change, finding rates of change, and the slope of curves.
• Integration: Deals with finding areas under curves and accumulated quantities.

In this context, differentiation is a crucial tool that allows us to determine how a function like \( y = e^{\frac{2x}{3}} \) changes as \( x \) changes. Calculus uses principles such as the chain rule to break down complex functions into manageable parts. This feature empowers students and professionals to predict trends and understand the dynamics of changing systems.

Differentiation

Differentiation is a process in calculus used to find the derivative of a function. The derivative measures how a function's output value changes with changes in its input value. It calculates the slope of the tangent line to the curve of the function at any given point.
Here are a few key points about differentiation:

• The derivative tells us the rate of change of the function.
• It can determine if the function is increasing, decreasing, or constant over a region.
• In practical terms, it's used in physics for velocity, in economics for cost functions, and more.

In the given exercise, differentiating \( y = e^{\frac{2x}{3}} \) involves finding the rate at which \( y \) changes with respect to \( x \). Using differentiation rules, such as the chain rule and properties of exponential functions, we find \( \frac{dy}{dx} = \frac{2}{3} e^{\frac{2x}{3}} \). This result shows precisely how \( y \) varies with \( x \), a core example of how differentiation gives insight into the behavior of functions.