Problem 18 - Thomas Calculus

Question

In Exercises \(17-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=e^{2 x / 3} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Given Function

The problem provides the function \( y = e^{2x/3} \). This is an exponential function where the exponent is a function of \( x \). We are tasked with finding its derivative.

Step 2: Recall the Derivative of the Exponential Function

For an exponential function of the form \( e^{u(x)} \), its derivative with respect to \( x \) is \( e^{u(x)} \cdot u'(x) \). Here, \( u(x) = \frac{2x}{3} \).

Step 3: Differentiate the Exponent

Differentiate the exponent \( u(x) = \frac{2x}{3} \) with respect to \( x \). This results in \( u'(x) = \frac{2}{3} \).

Step 4: Apply the Chain Rule

Using the chain rule, the derivative of \( y = e^{2x/3} \) with respect to \( x \) is given by \( \frac{dy}{dx} = e^{2x/3} \cdot \frac{2}{3} \).

Step 5: Simplify the Expression

Simplify the expression for the derivative. So \( \frac{dy}{dx} = \frac{2}{3} e^{2x/3} \).

Key Concepts

Exponential Function

An exponential function is a type of mathematical function that involves the constant base, usually denoted as \( e \), raised to a variable power. In exponential functions, the variable is placed in the exponent, such as in \( y = e^{2x/3} \). These functions are important in many fields because they describe how quantities change proportionally. They are commonly used in applications involving growth and decay, such as population growth, radioactive decay, and compound interest.

When dealing with exponential functions for differentiation purposes, it is crucial to recognize that the derivative of an exponential function of the form \( e^{u(x)} \) involves not just the original function itself, but also the derivative of the exponent. This leads us to the use of the chain rule in differentiation to solve such problems effectively.

Chain Rule

The chain rule is a fundamental differentiation technique used when finding the derivative of a composite function. If you have a function \( y = e^{u(x)} \), where \( u(x) \) is some function of \( x \), the chain rule is used to differentiate it. The chain rule states that the derivative of the composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In simpler terms:

• First, differentiate the outer function \( e^{u(x)} \) which is simply \( e^{u(x)} \).
• Then, multiply this by the derivative of the inner function \( u(x) \).

For the given function \( y = e^{2x/3} \), after identifying \( u(x) = 2x/3 \), we differentiate \( u(x) \), which gives us \( u'(x) = 2/3 \). Hence, by the chain rule, the derivative \( \frac{dy}{dx} \) is given by \( e^{2x/3} \times \frac{2}{3} \).

The chain rule is very useful because it allows differentiation in situations where functions are nested within each other, providing a straightforward way to handle complexity in equations.

Differentiation Process

The process of differentiation involves finding the derivative of a function, which is essentially the rate at which the function's value changes as its input changes. Differentiation is a key concept in calculus that allows us to understand and quantify change.

For the function \( y = e^{2x/3} \), the differentiation process involves several steps:

• Firstly, identify the type of function, which in this case is an exponential function with an exponent that is a linear function of \( x \).
• Next, recognize that the chain rule will be required to differentiate the function due to the composite nature of the exponent.
• Differentiating the inner function \( u(x) = 2x/3 \) yields \( u'(x) = 2/3 \).
• Finally, applying the chain rule gives \( \frac{dy}{dx} = \frac{2}{3} e^{2x/3} \), which is the derivative of the original function.

The differentiation process can transform both simple and complicated functions by breaking them into manageable parts and calculating how each part influences the change in the function's output. Developing a solid understanding of differentiation techniques like the chain rule is crucial for success in calculus and myriad applications beyond it.