Questions: Calculate the partial derivatives ∂U/∂T and ∂T/∂U using implicit differentiation of (TU-V)² ln(W-UV)=ln(13) at (T, U, V, W)=(3,4,13,65). (Use symbolic notation and fractions where needed.)

Calculate the partial derivatives ∂U/∂T and ∂T/∂U using implicit differentiation of (TU-V)² ln(W-UV)=ln(13) at (T, U, V, W)=(3,4,13,65). (Use symbolic notation and fractions where needed.)
Transcript text: Calculate the partial derivatives $\frac{\partial U}{\partial T}$ and $\frac{\partial T}{\partial U}$ using implicit differentiation of $(T U-V)^{2} \ln (W-U V)=\ln (13)$ at $(T, U, V, W)=(3,4,13,65)$. (Use symbolic notation and fractions where needed.)
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Solution

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Solution Steps

To find the partial derivatives \(\frac{\partial U}{\partial T}\) and \(\frac{\partial T}{\partial U}\), we will use implicit differentiation on the given equation \((T U - V)^2 \ln(W - U V) = \ln(13)\). First, differentiate both sides of the equation with respect to \(T\) and \(U\) separately, treating other variables as constants. Then, solve the resulting equations to find the desired partial derivatives at the given point \((T, U, V, W) = (3, 4, 13, 65)\).

Step 1: Define the Equation

We start with the equation given in the problem:

\[ (T U - V)^2 \ln(W - U V) = \ln(13) \]

Step 2: Differentiate with Respect to \(T\)

We differentiate the equation implicitly with respect to \(T\):

\[ \frac{\partial}{\partial T} \left( (T U - V)^2 \ln(W - U V) \right) = 0 \]

This results in:

\[ dU_dT_{eq} = 2U(TU - V) \ln(W - UV) \]

Step 3: Differentiate with Respect to \(U\)

Next, we differentiate the equation implicitly with respect to \(U\):

\[ \frac{\partial}{\partial U} \left( (T U - V)^2 \ln(W - U V) \right) = 0 \]

This results in:

\[ dT_dU_{eq} = 2T(TU - V) \ln(W - UV) - \frac{V(TU - V)^2}{W - UV} \]

Step 4: Evaluate at the Given Point

We evaluate both derivatives at the point \((T, U, V, W) = (3, 4, 13, 65)\):

  1. For \(dU_dT_{eq}\):

\[ dU_dT_{eq} = 2 \cdot 4 \cdot (3 \cdot 4 - 13) \ln(65 - 4 \cdot 13) = 2 \cdot 4 \cdot (12 - 13) \ln(65 - 52) = 2 \cdot 4 \cdot (-1) \ln(13) \]

  1. For \(dT_dU_{eq}\):

\[ dT_dU_{eq} = 2 \cdot 3 \cdot (3 \cdot 4 - 13) \ln(65 - 4 \cdot 13) - \frac{13(3 \cdot 4 - 13)^2}{65 - 4 \cdot 13} \]

Final Answer

Since both partial derivatives are not explicitly solvable from the equations derived, we conclude that:

\[ \frac{\partial U}{\partial T} = 0 \quad \text{and} \quad \frac{\partial T}{\partial U} = \text{undefined} \]

Thus, the boxed final answer is:

\[ \boxed{\frac{\partial U}{\partial T} = 0, \quad \frac{\partial T}{\partial U} = \text{undefined}} \]

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