https://file.aiquickdraw.com/mathbot/file/1729012676014ntfi5iej.jpg

We are tasked with differentiating the function:

\[
g(x) = \ln(x e^{-2x})
\]

### Step 1: Simplify the logarithmic expression
Using the logarithmic property \(\ln(a \cdot b) = \ln(a) + \ln(b)\), we can break down the given function:

\[
g(x) = \ln(x) + \ln(e^{-2x})
\]

Since \(\ln(e^{-2x}) = -2x\) (because \(\ln(e^a) = a\)):

\[
g(x) = \ln(x) - 2x
\]

### Step 2: Differentiate the simplified function
Now, we can differentiate each term separately:

1. The derivative of \(\ln(x)\) is \(\frac{1}{x}\).
2. The derivative of \(-2x\) is \(-2\).

Thus, the derivative of \(g(x)\) is:

\[
g'(x) = \frac{1}{x} - 2
\]

### Final Answer:
\[
g'(x) = \frac{1}{x} - 2
\]

Let me know if you need any further clarification or more details.

---

Here are 5 related questions you may want to explore:
1. How do you differentiate other logarithmic functions like \(\ln(x^2 + 1)\)?
2. What is the chain rule, and how is it used in differentiation?
3. Can you explain the derivative of exponential functions like \(e^{ax}\)?
4. What is the domain of the function \(g(x) = \ln(x e^{-2x})\)?
5. How do you apply the product rule in differentiation?

**Tip**: Always check if you can simplify logarithmic expressions before differentiating them—it makes the process much easier!

Learn how to differentiate the function g(x) = ln(x * e^(-2x)) with this step-by-step guide. The solution simplifies the logarithmic expression and applies the derivative rules for both ln(x) and exponential functions. Ideal for Grades 11-12 or early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation