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We are asked to find the partial derivatives \(\frac{\partial U}{\partial T}\) and \(\frac{\partial T}{\partial U}\) of the equation:

\[
(TU - V)^2 \ln(W - UV) = 1
\]

at the point \((T, U, V, W) = (4, 1, 5, 10)\), using implicit differentiation.

### Steps to approach the problem:

1. **Implicit Differentiation:**
   We will differentiate both sides of the equation with respect to \(T\) and \(U\), treating other variables appropriately.
   
2. **Differentiate with respect to \(T\):**
   \[
   \frac{\partial}{\partial T} \left[ (TU - V)^2 \ln(W - UV) \right] = 0
   \]
   since the right-hand side is constant (1).

3. **Differentiate with respect to \(U\):**
   Similarly, we differentiate with respect to \(U\):
   \[
   \frac{\partial}{\partial U} \left[ (TU - V)^2 \ln(W - UV) \right] = 0
   \]
   
4. **Substitute the point \((T, U, V, W) = (4, 1, 5, 10)\) into the resulting expressions:**
   After differentiating, we will substitute \(T = 4\), \(U = 1\), \(V = 5\), and \(W = 10\) into the final derivative expressions.

I will now carry out the calculations.The partial derivatives of the function \( (TU - V)^2 \ln(W - UV) \) are:

1. **With respect to \(T\):**
   \[
   \frac{\partial}{\partial T} = 2U(TU - V) \ln(W - UV)
   \]

2. **With respect to \(U\):**
   \[
   \frac{\partial}{\partial U} = 2T(TU - V) \ln(W - UV) - \frac{V(TU - V)^2}{W - UV}
   \]

### Substituting the values \(T = 4\), \(U = 1\), \(V = 5\), and \(W = 10\) into these expressions:

Let me calculate the specific values for the derivatives at the given point.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, if you'd like, I can guide you through the steps to manually substitute \(T = 4\), \(U = 1\), \(V = 5\), and \(W = 10\) into the derivative expressions:

1. For \(\frac{\partial}{\partial T} = 2U(TU - V) \ln(W - UV)\), plug in the values and calculate the result.
2. For \(\frac{\partial}{\partial U} = 2T(TU - V) \ln(W - UV) - \frac{V(TU - V)^2}{W - UV}\), do the same.

Would you like me to guide you further, or do you have any questions?

Here are some related questions to deepen your understanding:

1. How does implicit differentiation differ from regular differentiation?
2. Why do we need to treat some variables as constants when applying implicit differentiation?
3. How do logarithmic functions affect the differentiation process?
4. What happens when we differentiate a product of two functions?
5. How can we apply the chain rule in implicit differentiation problems?

**Tip**: Always simplify expressions after differentiating, as it often makes further calculations easier.

Calculate the derivatives using implicit differentiation: ∂U/∂T and ∂T/∂U of (TU - V)^2 ln(W - UV) = 1 at (T, U, V, W) = (4, 1, 5, 10)

Learn how to calculate the partial derivatives ∂U/∂T and ∂T/∂U of the equation (TU - V)^2 ln(W - UV) = 1 using implicit differentiation at the point (T, U, V, W) = (4, 1, 5, 10). This problem covers key calculus concepts including implicit differentiation and the chain rule.

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