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To find the derivative of the function 

\[ y = \frac{4 \ln(x+5)}{x^2}, \]

we'll use the quotient rule. The quotient rule states that if you have a function \( y = \frac{u(x)}{v(x)} \), then the derivative \( y' \) is given by:

\[ y' = \frac{u'v - uv'}{v^2}. \]

Here, \( u(x) = 4 \ln(x+5) \) and \( v(x) = x^2 \).

First, we find the derivatives \( u'(x) \) and \( v'(x) \):

1. \( u(x) = 4 \ln(x+5) \)
   \[ u'(x) = 4 \cdot \frac{1}{x+5} \cdot \frac{d}{dx}(x+5) = \frac{4}{x+5}. \]

2. \( v(x) = x^2 \)
   \[ v'(x) = 2x. \]

Now, applying the quotient rule:

\[ y' = \frac{\left(\frac{4}{x+5}\right) x^2 - (4 \ln(x+5)) (2x)}{(x^2)^2}. \]

Simplify the numerator and denominator:

\[ y' = \frac{4x^2}{x+5} - \frac{8x \ln(x+5)}{x^4}. \]

Combine the terms:

\[ y' = \frac{4x^2 - 8x(x+5)\ln(x+5)}{x^4(x+5)}. \]

Finally, simplify if possible:

\[ y' = \frac{4x - 8(x+5)\ln(x+5)}{x^3(x+5)}. \]

Thus, the derivative of the function \( y = \frac{4 \ln(x+5)}{x^2} \) is:

\[ y' = \frac{4x - 8(x+5)\ln(x+5)}{x^3(x+5)}. \]

Do you want further details or have any questions?

Here are some questions you might ask next:
1. How do you apply the product rule in calculus?
2. Can you explain the chain rule with an example?
3. What are common applications of derivatives in real life?
4. How do you find the second derivative of a function?
5. What is implicit differentiation and when is it used?

**Tip:** When working with logarithmic functions, remember the properties of logarithms to simplify derivatives and integrals.

Learn how to find the derivative of the function y = 4 ln(x+5) / x^2 using the quotient rule in calculus. This detailed solution covers the application of the quotient rule and simplification steps for logarithmic functions. Suitable for undergraduate students and anyone studying calculus.

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