https://file.aiquickdraw.com/mathbot/file/1729643589935zn6zdcro.jpg

The given problem asks:

**"If \( f(x) = \ln(3x^2 + x) \), find \( f'(1) \)."**

Let’s solve it step-by-step.

---

### Step 1: Differentiate \( f(x) = \ln(3x^2 + x) \) using the chain rule.
The chain rule states:
\[
\frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx}
\]
Here, \( u = 3x^2 + x \). So, first, differentiate \( u \) with respect to \( x \).

\[
\frac{du}{dx} = \frac{d}{dx}(3x^2 + x) = 6x + 1
\]

Now apply the chain rule to find \( f'(x) \):

\[
f'(x) = \frac{1}{3x^2 + x} \cdot (6x + 1)
\]

---

### Step 2: Evaluate \( f'(x) \) at \( x = 1 \).
Substitute \( x = 1 \) into the derivative expression:

\[
f'(1) = \frac{6(1) + 1}{3(1)^2 + 1} = \frac{6 + 1}{3 + 1} = \frac{7}{4}
\]

---

### Final Answer:
\[
f'(1) = \frac{7}{4}
\]

---

Would you like further details, or do you have other questions?

---

Here are five related questions:
1. How does the chain rule work in more complex functions like \( \ln(\sin(x^2)) \)?
2. What is the significance of evaluating a derivative at a specific point?
3. How would the derivative change if the argument inside the logarithm was \( 3x^2 + x + 1 \)?
4. What are the domain restrictions for the function \( f(x) = \ln(3x^2 + x) \)?
5. Can you apply implicit differentiation to logarithmic functions? 

**Tip:** Always ensure the argument of a logarithmic function is positive to avoid undefined values.

Learn how to differentiate the logarithmic function f(x) = ln(3x^2 + x) and find its derivative at x = 1 using the chain rule. This step-by-step solution covers key calculus concepts, suitable for undergraduate calculus 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