Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated derivative. \(D_{x} e^{x+2}\)

Step-by-Step Solution

Answer

The derivative of \( e^{x+2} \) is \( e^{x+2} \).

Step 1: Identify the Function

The given function is the exponential function: \( e^{x+2} \).

Step 2: Apply the Chain Rule

To find the derivative of \( e^{x+2} \), we need to use the chain rule. The chain rule states: if \( y = e^{u} \), where \( u \) is a function of \( x \), then \( \frac{dy}{dx} = \frac{du}{dx} \cdot e^{u} \). In this case, \( u = x+2 \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( u = x+2 \) with respect to \( x \). The derivative \( \frac{du}{dx} = \frac{d}{dx}(x+2) = 1 \).

Step 4: Calculate the Derivative Using Chain Rule

Apply the chain rule to find \( \frac{d}{dx}e^{x+2} \):\[ \frac{d}{dx} e^{x+2} = e^{x+2} \cdot 1 = e^{x+2} \].

Step 5: Write The Derivative

The derivative of \( e^{x+2} \) with respect to \( x \) is \( e^{x+2} \).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus, particularly when dealing with composite functions.
The principle behind the chain rule is fairly straightforward: it allows us to differentiate a composite function by focusing on its outer and inner components separately.

• Imagine a function inside another function, much like nesting dolls. The outer function is differentiated first, with the inner function kept intact.
• Then, differentiate the inner function, resulting in two derivatives that multiply each other.

This is exactly what we see in the given problem: the outer function is the exponential part, and the inner function is the exponent, namely, \( x+2 \).
By applying the chain rule:

• We first differentiate the outer function, \( e^{x+2} \), leaving \( x+2 \) unchanged, resulting in \( e^{x+2} \).
• Next, we find the derivative of the inner function \( x+2 \), which simplifies to \( 1 \).

Finally, multiplying these derivatives gives us the complete derivative for the composite function.

Exponential Function

The function we have, \( e^{x+2} \), is an example of an exponential function. Exponential functions are often recognized by having the variable as the exponent.
These functions exhibit unique properties that make them interesting to differentiate.

• For the base \( e \), which is approximately 2.718, the exponential function has a special property: its derivative is the same as the function itself.
• This is why when differentiating \( e^{x} \) or \( e^{x+2} \), the result remains \( e^{x+2} \).

This self-replicating nature simplifies differentiation greatly, especially when combined with the chain rule for more complex expressions like \( e^{x+2} \).
Understanding this feature is vital as it frequently appears in various mathematical and real-world applications, from growth modeling to decay processes.

Differentiation

Differentiation is a process in calculus used to determine the rate at which a function changes.
This concept is pivotal in understanding how functions behave as variables shift.For the function \( e^{x+2} \), applying differentiation means we're looking at how this specific expression changes as \( x \) changes.
Here's the step-by-step:

• Identify the function and recognize the need for a rule such as the chain rule when dealt with complex functions.
• Use the derivative of the basic exponential function, \( e^{x} \), which remains the same.
• In the case of the given function, since \( e^{x+2} \) is already adjusted for any constant shifts in \( x \), only the inner function \( x+2 \) requires differentiation, yielding \( 1 \).

Overall, differentiation provides invaluable insights into the behavior and characteristics of functions, allowing us to analyze and predict various aspects of mathematical functions.