What Is The Derivative Of Xe X

What is the Derivative of xeˣ? A Comprehensive Guide

The derivative of xeˣ is a classic example in calculus that demonstrates the power and application of the product rule. Understanding this derivative is crucial for mastering differential calculus and its applications in various fields like physics, engineering, and economics. This comprehensive guide will not only provide the solution but delve deep into the underlying principles, providing a thorough understanding of the process and its implications.

Understanding the Product Rule

Before we tackle the derivative of xeˣ, let's revisit the fundamental concept of the product rule. The product rule is essential for differentiating functions that are the product of two or more functions. It states:

If we have two differentiable functions, u(x) and v(x), then the derivative of their product is given by:

(d/dx)[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

In simpler terms, the derivative of a product is the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.

Applying the Product Rule to xeˣ

Now, let's apply the product rule to find the derivative of xeˣ. In this case:

• u(x) = x
• v(x) = eˣ

First, we find the derivatives of u(x) and v(x):

• u'(x) = d/dx (x) = 1 (The derivative of x with respect to x is 1)
• v'(x) = d/dx (eˣ) = eˣ (The derivative of eˣ with respect to x is eˣ)

Now, substitute these values into the product rule formula:

(d/dx)[xeˣ] = u'(x)v(x) + u(x)v'(x) = (1)(eˣ) + (x)(eˣ) = eˣ + xeˣ

Therefore, the derivative of xeˣ is eˣ + xeˣ. This can be further simplified by factoring out eˣ:

(d/dx)[xeˣ] = eˣ(1 + x)

Understanding the Result: A Deeper Dive

The derivative, eˣ(1 + x), reveals interesting properties and applications. Let's explore these further:

1. The Exponential Function's Dominance

Notice that the exponential function, eˣ, is a factor in the derivative. This highlights the dominance of exponential growth. No matter the value of x, the exponential term will significantly influence the overall rate of change.

2. Linear and Exponential Contributions

The derivative consists of two terms: eˣ and xeˣ. The term eˣ represents the contribution of the exponential function itself, while xeˣ represents the combined contribution of the linear function (x) and the exponential function. This illustrates how the interaction between different functions affects their overall derivative.

3. Applications in Differential Equations

The derivative of xeˣ often appears in the context of solving differential equations. These equations describe relationships between a function and its derivatives, and understanding how to differentiate expressions like xeˣ is fundamental to solving many important differential equations in various fields, such as:

• Physics: Modeling radioactive decay, population growth, and heat transfer.
• Engineering: Analyzing electrical circuits, mechanical systems, and chemical reactions.
• Economics: Describing economic growth and decay models.

Higher-Order Derivatives

We can extend this analysis to find higher-order derivatives of xeˣ. For instance, let's find the second derivative:

First derivative: f'(x) = eˣ + xeˣ = eˣ(1 + x)

Second derivative: f''(x) = d/dx [eˣ(1 + x)]

We again apply the product rule, with u(x) = eˣ and v(x) = (1 + x):

• u'(x) = eˣ
• v'(x) = 1

Therefore:

f''(x) = eˣ(1 + x) + eˣ(1) = eˣ + xeˣ + eˣ = 2eˣ + xeˣ = eˣ(2 + x)

Similarly, we can find the third derivative and so on. Each successive derivative will involve the exponential function and a polynomial term of increasing degree.

Practical Examples & Applications

Let's consider some practical scenarios where the derivative of xeˣ comes into play:

Example 1: Optimization Problems

Suppose we have a cost function C(x) = xeˣ, where x represents the number of units produced. To find the production level that minimizes the marginal cost (the rate of change of cost with respect to production), we would find the derivative, set it to zero, and solve for x.

C'(x) = eˣ(1 + x) = 0

This equation has only one solution, x = -1. However, since we can't produce a negative number of units, this solution is not physically meaningful in this context. The analysis reveals insights into the cost function's behavior even without a physically meaningful minimum.

Example 2: Modeling Growth

In population dynamics, the function P(t) = teᵗ could model a population's growth over time (t). The derivative, dP/dt = eᵗ(1 + t), represents the rate of population growth. Analyzing this derivative allows us to understand the speed at which the population is growing at any given time.

Example 3: Physics – Radioactive Decay with Production

In certain nuclear processes, the rate of change of the amount of a radioactive substance could be described by a differential equation involving a term like xeˣ, reflecting both decay and simultaneous production.

Conclusion

The derivative of xeˣ, which is eˣ(1+x), is more than just a simple calculation; it's a gateway to understanding the interplay between exponential and linear functions and their profound impact on various mathematical and real-world applications. This in-depth explanation not only provides the solution but emphasizes the underlying principles and practical uses of this crucial derivative, highlighting its significance in fields ranging from physics and engineering to economics and beyond. Understanding this derivative enhances your comprehension of differential calculus and allows for a more profound understanding of the mathematical models that describe our world.