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!