In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = xe^x-e^x

Verified step by step guidance

1. Identify the function given: \(y = xe^x - e^x\).
2. Recognize that the function is composed of two terms: \(xe^x\) and \(-e^x\), and you will need to differentiate each term separately.
3. For the first term \(xe^x\), apply the product rule for differentiation, which states: \(\frac{d}{dx}[u v] = u' v + u v'\). Here, let \(u = x\) and \(v = e^x\).
4. Calculate the derivatives of \(u\) and \(v\): \(u' = \frac{d}{dx}[x] = 1\) and \(v' = \frac{d}{dx}[e^x] = e^x\).
5. Apply the product rule to the first term: \(\frac{d}{dx}[xe^x] = 1 \cdot e^x + x \cdot e^x = e^x + xe^x\). Then differentiate the second term \(-e^x\) as \(-e^x\). Finally, combine these results to write the derivative of \(y\).

Key Concepts

Derivative of Exponential Functions

The derivative of the exponential function e^x is e^x itself. This property is fundamental when differentiating expressions involving e^x, as it simplifies the process and helps in applying rules like the product and chain rules effectively.

Product Rule

The product rule is used to differentiate functions that are products of two differentiable functions. It states that the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x). This rule is essential for differentiating terms like xe^x.

Basic Differentiation Rules

Basic differentiation rules include the power rule, constant multiple rule, and sum/difference rule. These rules allow us to differentiate terms like x and combine derivatives of multiple terms, such as in y = xe^x - e^x.