Find d/dx (In(xe^x)) without using the Chain Rule and the Product Rule.

Verified step by step guidance

1. Step 1: Begin by recognizing that the expression inside the logarithm is a product: \(xe^x\). We can use the property of logarithms that \(\ln(a \cdot b) = \ln(a) + \ln(b)\). Apply this property to rewrite the expression as \(\ln(x) + \ln(e^x)\).
2. Step 2: Simplify \(\ln(e^x)\). Recall that \(\ln(e^x) = x \cdot \ln(e)\), and since \(\ln(e) = 1\), this simplifies to \(x\). Therefore, the expression becomes \(\ln(x) + x\).
3. Step 3: Differentiate \(\ln(x)\) with respect to \(x\). The derivative of \(\ln(x)\) is \(\frac{1}{x}\).
4. Step 4: Differentiate \(x\) with respect to \(x\). The derivative of \(x\) is simply \(1\).
5. Step 5: Combine the derivatives from Step 3 and Step 4. The derivative of \(\ln(x) + x\) is \(\frac{1}{x} + 1\).

Key Concepts

Natural Logarithm Properties

Understanding the properties of natural logarithms is essential for simplifying expressions involving ln. Specifically, the property ln(a*b) = ln(a) + ln(b) allows us to break down the logarithm of a product into a sum, which can simplify differentiation.

Exponential Functions

Exponential functions, such as e^x, have unique properties that make them easier to differentiate. The derivative of e^x is e^x itself, which simplifies calculations significantly when combined with logarithmic functions.

Basic Differentiation Rules

Familiarity with basic differentiation rules, such as the derivative of a constant and the power rule, is crucial. These rules provide the foundation for finding derivatives without relying on more complex techniques like the Chain Rule or Product Rule.