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Finding the Derivative of x²eˣ: A Comprehensive Guide

Finding the derivative of functions involving the product of a polynomial and an exponential function, like x²eˣ, requires a fundamental understanding of calculus. This article will provide a step-by-step guide on how to derive this function, explore the underlying mathematical principles, and delve into some related applications. We'll also address common questions and misconceptions. This comprehensive approach aims to not only give you the answer but also a deeper understanding of the process. Understanding derivatives is crucial in various fields, including physics, engineering, and economics, where rates of change are vital.

Introduction: Understanding Derivatives

Before we dive into the specifics of deriving x²eˣ, let's briefly revisit the concept of a derivative. In calculus, the derivative of a function represents its instantaneous rate of change at any given point. Geometrically, it represents the slope of the tangent line to the function's graph at that point. The process of finding a derivative is called differentiation.

Several rules govern differentiation. For this particular problem, we'll primarily use the product rule and the knowledge of derivatives of common functions like polynomials and exponentials.

The Product Rule: A Cornerstone of Differentiation

The product rule is essential when dealing with functions that are the product of two or more simpler functions. It states:

If we have a function f(x) = u(x) * v(x), then its derivative is given by:

f'(x) = u'(x)v(x) + u(x)v'(x)

In simpler terms: The derivative of a product is the derivative of the first function times the second function, plus the first function times the derivative of the second function.

Step-by-Step Derivation of x²eˣ

Now, let's apply this knowledge to find the derivative of x²eˣ. In this case:

u(x) = x² v(x) = eˣ

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

u'(x) = 2x (This is the derivative of x², using the power rule) v'(x) = eˣ (This is the derivative of eˣ, a fundamental result in calculus)

Now, we apply the product rule:

f'(x) = u'(x)v(x) + u(x)v'(x)

Substituting our values:

f'(x) = (2x)(eˣ) + (x²)(eˣ)

We can factor out eˣ to simplify the expression:

f'(x) = eˣ(2x + x²)

Therefore, the derivative of x²eˣ is eˣ(2x + x²).

Explanation with Detailed Mathematical Justification

The derivation above relies on several key concepts from calculus:

Power Rule: The power rule states that the derivative of xⁿ is nxⁿ⁻¹. This is how we obtained the derivative of x² as 2x.
Derivative of eˣ: The derivative of the exponential function eˣ is remarkably simple: it's just eˣ itself. This is a fundamental property of the exponential function. This arises from the definition of e as the base of the natural logarithm, which is intrinsically linked to the exponential function's rate of change being equal to itself.
Product Rule Justification: The product rule's validity can be proven using the limit definition of the derivative and algebraic manipulation. The proof involves expressing the change in the product function as a sum of changes involving both functions. Taking the limit as the change in x approaches zero yields the product rule formula. The proof is rigorous but beyond the scope of this introductory guide.
Linearity of Differentiation: Note that the derivative operator is linear. This means that the derivative of a sum is the sum of the derivatives, and a constant multiplier can be pulled out of the derivative. This property was implicitly used when we factored out eˣ in the final simplification.

Exploring Further: Higher-Order Derivatives

The process doesn't stop at the first derivative. We can find higher-order derivatives by repeatedly applying the differentiation process. Let's find the second derivative of x²eˣ:

We already have the first derivative: f'(x) = eˣ(2x + x²)

To find the second derivative (f''(x)), we again apply the product rule:

Let u(x) = eˣ and v(x) = 2x + x². Then:

u'(x) = eˣ v'(x) = 2 + 2x

Applying the product rule:

f''(x) = u'(x)v(x) + u(x)v'(x) = eˣ(2x + x²) + eˣ(2 + 2x)

Simplifying:

f''(x) = eˣ(2x + x² + 2 + 2x) = eˣ(x² + 4x + 2)

Therefore, the second derivative of x²eˣ is eˣ(x² + 4x + 2). Similarly, we can continue this process to find third, fourth, and higher-order derivatives.

Applications of Derivatives: Real-World Examples

The ability to find derivatives, and specifically the derivative of functions like x²eˣ, has far-reaching applications across numerous fields:

Physics: Derivatives are crucial in understanding motion and change. For example, the velocity of an object is the derivative of its position with respect to time, and acceleration is the derivative of velocity. Complex motions might involve functions similar to x²eˣ, and understanding their derivatives is critical for analyzing the motion.
Engineering: In engineering design and analysis, derivatives are used to optimize designs, model dynamic systems, and solve differential equations. For instance, analyzing the rate of heat transfer or the stress on a structure might involve functions whose derivatives are essential for understanding the system's behavior.
Economics: Derivatives play a vital role in economic modeling. For example, marginal cost (the cost of producing one more unit) is the derivative of the total cost function. Analyzing the rate of change of profits, revenues, or other economic variables often requires differentiation.
Machine Learning: Derivatives form the backbone of many machine learning algorithms. Gradient descent, a fundamental optimization technique, relies on calculating the derivatives of a cost function to find its minimum value. This is used extensively to train models in tasks like image recognition, natural language processing, and many more.

Frequently Asked Questions (FAQ)

Q: What if the exponential term was e⁻ˣ instead of eˣ?

A: The process remains the same. You would still apply the product rule, but the derivative of e⁻ˣ is -e⁻ˣ. The final answer would reflect this change in sign.

Q: Can I use other differentiation techniques?

A: While the product rule is the most straightforward approach for x²eˣ, more advanced techniques like logarithmic differentiation might be applicable in some more complex scenarios involving products and exponential functions. However, for this specific problem, the product rule is efficient and clear.

Q: What happens if the polynomial term is more complex?

A: The product rule still applies. If you had, for instance, (x³ + 2x - 5)eˣ, you would find the derivative of the polynomial (3x² + 2) and then apply the product rule in the same manner as shown above.

Q: Why is the derivative of eˣ equal to eˣ?

A: This is a fundamental property of the exponential function eˣ. It stems from the definition of e as the limit of (1 + 1/n)ⁿ as n approaches infinity, and its connection to the natural logarithm. The proof of this property requires a deeper understanding of limits and the properties of the exponential and logarithmic functions.

Conclusion: Mastering Derivatives for Future Success

Understanding derivatives is a cornerstone of calculus and has wide-ranging applications in numerous fields. This detailed guide has provided a step-by-step explanation of finding the derivative of x²eˣ, emphasizing the importance of the product rule and providing context for its application. By understanding the mathematical underpinnings and exploring real-world examples, you've gained a deeper comprehension of this crucial concept. Mastering derivative calculations equips you with a valuable tool for tackling complex problems and gaining a deeper understanding of the world around us. Remember that practice is key to solidifying your understanding of these concepts. Continuously working through examples and challenging yourself with more complex problems will build your proficiency in calculus.