Problem 54 - Calculus Early Transcendentals

Question

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(x^{e}+e^{x}\right)$$

Step-by-Step Solution

Answer

Question: Find the derivative of the function \(f(x) = x^e + e^x\) with respect to \(x\). Answer: The derivative of the function \(f(x) = x^e + e^x\) with respect to \(x\) is \(f'(x) = ex^{e-1} + e^x\).

Step 1: Differentiate the first term

To differentiate the first term, we will use the power rule: \(\frac{d}{d x}(x^n) = nx^{n-1}\). Here, \(n = e\) that is the constant \(e\). So, we have: \(\frac{d}{d x}(x^e) = ex^{e-1}\)

Step 2: Differentiate the second term

To differentiate the second term, we will use the exponential rule: \(\frac{d}{d x}(e^x) = e^x\). We have: \(\frac{d}{d x}(e^x) = e^x\)

Step 3: Combine the derivatives

Now let's combine the derivatives we found in Steps 1 and 2 using the sum rule. The sum rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. In this case, the two functions are \(x^e\) and \(e^x\). So, we have: \(\frac{d}{d x}\left(x^{e}+e^{x}\right) = \frac{d}{d x}(x^e) + \frac{d}{d x}(e^x)\) Then, substitute the derivatives we've found: \(\frac{d}{d x}\left(x^{e}+e^{x}\right) = ex^{e-1} + e^x\)

Key Concepts

Power Rule

The Power Rule is one of the most essential tools in calculus for finding derivatives. It provides a straightforward way to differentiate functions of the form \(x^n\), where \(n\) can be any real number.
A simple way to remember the Power Rule is: take the exponent \(n\) of the term \(x^n\), multiply it by the coefficient (if there is one), and then decrease the exponent by one. Mathematically, the rule is expressed as:

• \(\frac{d}{dx}(x^n) = nx^{n-1}\)

In our exercise, the term \(x^e\) features a special constant \(e\) as its exponent. The process, however, remains the same. You differentiate by multiplying with \(e\) and then reducing the exponent by one, giving us \(ex^{e-1}\). This showcases the versatility of the Power Rule, even when exponents are constants like \(e\), the approximate value of 2.718. Remember, this rule assists with any polynomial-like term, making it a must-have in your calculus toolkit.

Exponential Rule

The Exponential Rule simplifies the differentiation of exponential functions that have the base \(e\), which is a naturally occurring constant approximately equal to 2.718.
Exponential functions are unique because the function's derivative is proportional to the function itself. The rule can be formulated as:

• \(\frac{d}{dx}(e^x) = e^x\)

This means that when you take the derivative of \(e^x\), it remains unchanged. This self-replicating characteristic is exclusive to \(e^x\) and doesn't hold for exponential functions with different bases.
In our exercise, we apply this rule directly to \(e^x\) to get its derivative, which remains \(e^x\). This relatively uncomplicated process is what makes the Exponential Rule one of the favorites among students tackling calculus.

Sum Rule

Understanding the Sum Rule is crucial when dealing with functions that consist of the sum of different parts, like the one in our exercise.
The Sum Rule states that the derivative of a sum of functions is the sum of their individual derivatives. This decomposition simplifies complex derivatives by allowing you to focus on each part separately. It is expressed as:

• \(\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))\)

In the given problem, we used the Sum Rule to tackle \(x^e + e^x\). By distinguishing the function into two separate parts, \(x^e\) and \(e^x\), we effortlessly apply the derivatives we found using the Power Rule and Exponential Rule, respectively.
This principle is widely applied in many areas of mathematics, making it indispensable for simplifying and solving differential equations and various calculus problems.