15–48. Derivatives Find the derivative of the following functions.
y = e^x x^e

Verified step by step guidance

1. Step 1: Identify the function y = e^x * x^e. This is a product of two functions, e^x and x^e, so we will use the product rule to find the derivative.
2. Step 2: Recall the product rule for derivatives, which states that if you have a function y = u * v, then the derivative y' = u' * v + u * v'.
3. Step 3: Differentiate the first function u = e^x. The derivative of e^x with respect to x is e^x.
4. Step 4: Differentiate the second function v = x^e. Use the power rule for derivatives, which states that the derivative of x^n is n * x^(n-1). Here, n = e, so the derivative is e * x^(e-1).
5. Step 5: Apply the product rule using the derivatives found: y' = (e^x) * (x^e) + (e^x) * (e * x^(e-1)). Simplify the expression to get the final derivative.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and chain rule, depending on the form of the function.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, such as in the given function y = e^x x^e.

Exponential Functions

Exponential functions are functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. The derivative of an exponential function, particularly when the base is e (Euler's number), is unique because it equals the function itself, i.e., d/dx(e^x) = e^x. Understanding how to differentiate exponential functions is crucial for solving problems involving them, such as the function in the question.