Find the derivative of the function.
yequals
StartFraction 2 ln left parenthesis x plus 8 right parenthesis Over x squared EndFraction
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Part 1
StartFraction dy Over dx EndFraction
equalsStartFraction 2 x minus 4 x cubed ln left parenthesis x plus 8 right parenthesis Over x Superscript 4 Baseline left parenthesis x plus 8 right parenthesis EndFraction

To find the derivative of the function

\[
y = \frac{2 \ln(x + 8)}{x^2}
\]

we will use the **quotient rule** for derivatives and the **chain rule** for the logarithmic function. The quotient rule states:

\[
\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\]

where \( u = 2 \ln(x + 8) \) and \( v = x^2 \).

### Step 1: Differentiate \( u = 2 \ln(x + 8) \)

Using the chain rule, we differentiate \( u \):

\[
\frac{du}{dx} = 2 \cdot \frac{1}{x + 8} \cdot 1 = \frac{2}{x + 8}
\]

### Step 2: Differentiate \( v = x^2 \)

\[
\frac{dv}{dx} = 2x
\]

### Step 3: Apply the quotient rule

Now, apply the quotient rule:

\[
\frac{dy}{dx} = \frac{x^2 \cdot \frac{2}{x + 8} - 2 \ln(x + 8) \cdot 2x}{(x^2)^2}
\]

### Step 4: Simplify the expression

First, simplify the numerator:

\[
x^2 \cdot \frac{2}{x + 8} = \frac{2x^2}{x + 8}
\]

And the second term:

\[
2 \ln(x + 8) \cdot 2x = 4x \ln(x + 8)
\]

So the numerator becomes:

\[
\frac{2x^2}{x + 8} - 4x \ln(x + 8)
\]

Now, for the denominator:

\[
(x^2)^2 = x^4
\]

Thus, the derivative is:

\[
\frac{dy}{dx} = \frac{\frac{2x^2}{x + 8} - 4x \ln(x + 8)}{x^4}
\]

### Step 5: Further simplify

We can combine terms in the numerator:

\[
\frac{dy}{dx} = \frac{2x - 4x^3 \ln(x + 8)}{x^4 (x + 8)}
\]

This is the final derivative.

\[
\frac{dy}{dx} = \frac{2x - 4x^3 \ln(x + 8)}{x^4 (x + 8)}
\]

---

Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you apply the product rule for differentiation?
2. What is the chain rule in calculus, and when is it useful?
3. How can we simplify logarithmic expressions before differentiating?
4. What is the derivative of a natural logarithmic function like \( \ln(x) \)?
5. How does the quotient rule compare with the product rule in terms of complexity?

**Tip:** Always ensure that terms in both the numerator and denominator are fully simplified before performing the final division.

Find the derivative of the function: y = (2 ln(x + 8)) / (x^2)

Explore how to find the derivative of the function y = (2 ln(x + 8)) / (x^2) using the quotient and chain rules. This calculus problem covers logarithmic differentiation and provides a step-by-step breakdown. Suitable for students in Grades 11-12 or college-level calculus.

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