Problem 12 - Technical Mathematics with Calculus

Question

Derivative of \(e^{u}\). Differentiate. $$y=x e^{x}$$

Step-by-Step Solution

Answer

The derivative of y = xe^{x} is y' = e^{x} + xe^{x}.

Step 1: Recognize the product of functions

The given function, y = xe^{x}, is a product of two functions: f(x) = x and g(x) = e^{x}. Therefore, we need to use the product rule for differentiation.

Step 2: Use the product rule for differentiation

The product rule states that if y = f(x)g(x), then the derivative y' = f'(x)g(x) + f(x)g'(x). So, differentiate f(x) = x to get f'(x) = 1 and differentiate g(x) = e^{x} to get g'(x) = e^{x}.

Step 3: Apply the product rule

Substitute f'(x), g(x), and g'(x) into the product rule formula: y' = (1)(e^{x}) + (x)(e^{x}) = e^{x} + xe^{x}.

Key Concepts

Derivative of Exponential Functions

Understanding the derivative of exponential functions is a key concept in calculus, particularly when dealing with natural exponential functions of the form e^x. The natural exponential function is special because its derivative is the same as the original function. That is, if you have a function f(x) = e^u, where u is a function of x, the derivative f'(x) is u'e^u. This property makes e (Euler's number) a fundamental constant in mathematics.

In the given exercise, we differentiate y = xe^x, which can seem complex at first. It's essential to recognize that e^x will remain unchanged when differentiated, and if u is a function of x, such as our x in xe^x, we’ll use the chain rule to find the derivative of that component. We must be precise and understand that the derivative of e^x is straightforward but requires careful application when combined with other functions.

Chain Rule

The chain rule is a powerful tool used when differentiating compositions of functions or functions of functions. It states that if you have a composite function f(g(x)), the derivative f'(x) is g'(x) multiplied by the derivative of f with respect to g(x), denoted as f'(g(x)).

Consider the function u(x) = g(x) and that e^u is our f(u). To find the derivative of e^u, we need to take the derivative of e^u with respect to u, which is e^u itself, and then multiply it by the derivative of u with respect to x, or u'(x). This application of the chain rule allows us to handle more complex derivatives of exponential functions effectively.

Calculus Applications

Calculus has a broad range of applications across various fields, and understanding how to work with derivatives is crucial for solving real-world problems. The derivative represents the rate at which a function is changing at any given point and is foundational to the fields of physics, engineering, economics, and beyond.

For instance, in physics, the derivative of position with respect to time gives us the velocity, and the derivative of velocity gives us acceleration. In the context of our exercise, the derivative of y = xe^x can represent numerous physical situations, such as continuous growth models where a quantity grows at a rate proportional to its current value multiplied by another varying factor. This fundamental concept helps students to apply mathematical theories to tangible scenarios, thereby enhancing their understanding and appreciation of calculus.