Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} e^{x+2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Function

The function given is an exponential function of the form \( e^{u(x)} \), where \( u(x) = x + 2 \).

Step 2: Recall the Derivative Rule for Exponential Functions

For a function \( e^{u(x)} \), the derivative with respect to \( x \) is \( u'(x) \, e^{u(x)} \).

Step 3: Differentiate the Exponent Function

Compute the derivative of \( u(x) = x + 2 \) with respect to \( x \). The derivative is \( u'(x) = 1 \) since the derivative of \( x \) is 1 and the derivative of a constant (2) is 0.

Step 4: Apply the Derivative Rule

Substitute \( u'(x) = 1 \) and the original function into the derivative rule: \[ \frac{d}{dx} e^{x+2} = 1 \cdot e^{x+2} \].

Step 5: Simplify the Expression

Simplify the expression obtained: \( e^{x+2} \), which is the derivative of the given function with respect to \( x \).

Key Concepts

Exponential Functions

Exponential functions are a key class of functions in mathematics, especially when studying calculus. They are expressed in the form \( e^{u(x)} \), where \( e \) is Euler's number, approximately 2.718, a fundamental constant in mathematics. In exponential functions, the variable \( x \) appears in the exponent, as opposed to polynomial functions, where \( x \) is in the base. This unique feature leads to some intriguing properties and applications, particularly in growth and decay processes.
Common examples of exponential functions include natural exponential \( e^x \), and they often model real-world phenomena like population growth, radioactive decay, and compound interest. These functions grow or decay at rates proportional to their current value, making them highly predictable over time.

• The exponential growth occurs when the exponent \( x \) increases, resulting in the base \( e \) being raised to a larger power.
• Exponential decay, on the other hand, occurs when \( x \) decreases, causing the overall value to reduce.

Understanding these functions helps grasp how different systems evolve and respond to changes over time, making them indispensable in both theoretical and applied contexts.

Calculus Differentiation

Calculus differentiation is a process that helps us understand how functions change. It provides a way to calculate the rate at which values in a function are changing at any given point. It's a fundamental tool in calculus, used to solve a variety of real-world problems.
With differentiation, we can figure out the slope of the tangent line to a curve at any point. Imagine you're driving along a road (the curve) and want to know the steepness of your path. Differentiation tells you exactly that slope.
The process involves several rules and techniques, such as the product rule, quotient rule, and chain rule, which are essential when dealing with more complex functions.

• The **product rule** is used when differentiating products of two functions.
• The **quotient rule** applies when you have one function divided by another.
• The **chain rule** is particularly important when differentiating compositions of functions, like exponential functions.

By practicing differentiation, students gain the skills needed to tackle various mathematical challenges, especially those involving rates of change, optimization problems, and motion dynamics.

Derivative Rules

Derivative rules are the systematic processes used to find derivatives, providing a shortcut to differentiation. These rules are crucial for efficiently solving calculus problems and are widely applied in both simple and complex scenarios.
One of the most straightforward rules is the derivative of power functions. For a power function \( f(x) = x^n \), the derivative is given by \( nf(x)^{n-1} \).
For exponential functions like \( e^{u(x)} \), the chain rule comes into play. The derivative of an exponential function \( e^{u(x)} \) with respect to \( x \) is \( u'(x) \, e^{u(x)} \). This rule combines the simplicity of exponential growth with the complexity of the chain rule:

• First, find the derivative of the exponent \( u(x) \), noted as \( u'(x) \).
• Then multiply \( u'(x) \) by the original exponential function \( e^{u(x)} \).

This approach is used in the given exercise, where the function \( e^{x+2} \) was differentiated. The solution followed these rules to arrive at the derivative \( e^{x+2} \), emphasizing the power of these derivative rules in simplifying calculus tasks. Understanding these principles is key to mastering calculus and applying it to solve real-world problems.