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Derive the following expression with descriptive solution in mathematical language

The given expression is:

\[
y = \ln \left( x^3 \cdot (x + 4)^2 \right)
\]

To differentiate this expression with respect to \( x \), we'll use the following steps:

### Step 1: Apply the Logarithm Property
Recall that the logarithm of a product is the sum of the logarithms:

\[
\ln(a \cdot b) = \ln(a) + \ln(b)
\]

So, we can rewrite the expression as:

\[
y = \ln(x^3) + \ln\left((x + 4)^2\right)
\]

### Step 2: Differentiate Each Term

Now, we differentiate each term with respect to \( x \).

#### Differentiating \( \ln(x^3) \):
\[
\frac{d}{dx} \ln(x^3) = \frac{d}{dx} \left( 3 \ln(x) \right) = 3 \cdot \frac{1}{x} = \frac{3}{x}
\]

#### Differentiating \( \ln\left((x + 4)^2\right) \):
\[
\frac{d}{dx} \ln\left((x + 4)^2\right) = \frac{d}{dx} \left( 2 \ln(x + 4) \right) = 2 \cdot \frac{1}{x + 4} \cdot \frac{d}{dx}(x + 4) = \frac{2}{x + 4}
\]

### Step 3: Combine the Results

Adding the derivatives of the two terms, we get the derivative of \( y \) with respect to \( x \):

\[
\frac{dy}{dx} = \frac{3}{x} + \frac{2}{x + 4}
\]

Thus, the derivative of the given function is:

\[
\frac{dy}{dx} = \frac{3}{x} + \frac{2}{x + 4}
\]

Would you like more details on any part of this process or have further questions? 

Here are five related questions you might find useful:

1. How does the chain rule apply in logarithmic differentiation?
2. What happens if the base of the logarithm is different from \( e \)?
3. How would the derivative change if the exponent on \( x \) was not 3 but another number?
4. Can we simplify the derivative further if we combine the fractions?
5. How would you find the second derivative of the given function?

**Tip:** Remember that logarithmic differentiation is particularly useful when dealing with products, quotients, or powers involving variables, as it can simplify the differentiation process significantly.

Learn how to differentiate ln(x^3 * (x + 4)^2) with a step-by-step solution. This problem covers logarithmic differentiation, differentiation rules, and the chain rule. Suitable for advanced high school and college students.

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