R(t)= 6000/
(16+t squared)( t/4+1)

​
 − 48000tln( t/4+1)/
(16+tsquared) sqaured

The function you've provided appears to involve the rate of change of a quantity with respect to time \( t \). It seems to have two components. I'll rewrite it in a more standard form to help clarify the expression:

\[
R(t) = \frac{6000}{(16 + t^2)\left(\frac{t}{4} + 1\right)} - \frac{48000t \ln\left(\frac{t}{4} + 1\right)}{(16 + t^2)^2}
\]

### Key Components:
1. **First Term:**
   \[
   \frac{6000}{(16 + t^2)\left(\frac{t}{4} + 1\right)}
   \]
   - This term is a fraction where the numerator is 6000 and the denominator is a product of two factors: \(16 + t^2\) and \(\frac{t}{4} + 1\).

2. **Second Term:**
   \[
   - \frac{48000t \ln\left(\frac{t}{4} + 1\right)}{(16 + t^2)^2}
   \]
   - This term involves a logarithmic function \(\ln\left(\frac{t}{4} + 1\right)\), multiplied by \(t\), and divided by the square of \(16 + t^2\).

### Further Steps:
If you need to analyze this function further, such as finding its derivative, integral, or evaluating it at specific values of \(t\), please specify. Alternatively, I can help you simplify the expression or solve specific related problems.

Would you like more details, or do you have any specific questions? Here are some potential follow-up questions:

1. Would you like to simplify or factorize any part of the function?
2. Do you need to find the derivative of \(R(t)\)?
3. Are you interested in finding the critical points of \(R(t)\)?
4. Should we evaluate \(R(t)\) for specific values of \(t\)?
5. Would you like to plot the function to visualize its behavior?

### Tip:
When working with complex expressions, breaking them down into smaller components can make it easier to analyze and solve.

Explore the function R(t) = 6000 / ((16 + t^2)(t/4 + 1)) - 48000t ln(t/4 + 1) / (16 + t^2)^2 using calculus techniques. Understand its components, derivative, and applications. Ideal for advanced mathematics students and professionals.

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