To find the derivative of \( \ln(x^4 + x^3) \), we can apply the chain rule. Here's the process:

Given function:
\[
y = \ln(x^4 + x^3)
\]

### Step 1: Differentiate the outer function
The derivative of \( \ln(u) \) with respect to \( u \) is:
\[
\frac{d}{du} \ln(u) = \frac{1}{u}
\]
So, the derivative of \( \ln(x^4 + x^3) \) is:
\[
\frac{1}{x^4 + x^3}
\]

### Step 2: Differentiate the inner function
Now, differentiate the inner function \( x^4 + x^3 \) with respect to \( x \):
\[
\frac{d}{dx}(x^4 + x^3) = 4x^3 + 3x^2
\]

### Step 3: Apply the chain rule
Now, multiply the derivative of the outer function by the derivative of the inner function:
\[
\frac{d}{dx} \ln(x^4 + x^3) = \frac{1}{x^4 + x^3} \cdot (4x^3 + 3x^2)
\]

Thus, the derivative is:
\[
\frac{d}{dx} \ln(x^4 + x^3) = \frac{4x^3 + 3x^2}{x^4 + x^3}
\]

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. What is the derivative of \( \ln(x^2 + 1) \)?
2. How do you differentiate \( \ln(ax + b) \) where \( a \) and \( b \) are constants?
3. What is the chain rule, and how is it applied in different scenarios?
4. How would you differentiate a product of logarithmic and polynomial functions?
5. Can you generalize the derivative of \( \ln(f(x)) \) for any function \( f(x) \)?

**Tip:** Always remember to apply the chain rule when differentiating composite functions!

Learn how to find the derivative of ln(x^4 + x^3) step by step using the chain rule. This problem highlights important calculus concepts such as logarithmic differentiation and chain rule application. Suitable for students in Grades 11-12 and early college calculus courses.

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