Calculate the partial derivatives and using implicit differentiation of (TU – V)² In (W – UV) = ln (6) at (T, U, V, W) = (1,5,6, 36). au (Use symbolic notation and fractions where needed.) JU ƏT ƏT JU =
Calculate the partial derivatives and using implicit differentiation of (TU – V)² In (W – UV) = ln (6) at (T, U, V, W) = (1,5,6, 36). au (Use symbolic notation and fractions where needed.) JU ƏT ƏT JU =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Implicit Differentiation and Partial Derivatives
#### Problem Statement
Calculate the partial derivatives \( \frac{\partial U}{\partial T} \) and \( \frac{\partial T}{\partial U} \) using implicit differentiation of \((TU - V)^2 \ln(W - UV) = \ln(6)\) at \((T, U, V, W) = (1, 5, 6, 36)\).
(Use symbolic notation and fractions where needed.)
#### Solution
1. **First Partial Derivative \(\frac{\partial U}{\partial T}\)**
\[ \frac{\partial U}{\partial T} = \]
2. **Second Partial Derivative \(\frac{\partial T}{\partial U}\)**
\[ \frac{\partial T}{\partial U} = \]
#### Explanation
Given the equation \((TU - V)^2 \ln(W - UV) = \ln(6)\), we need to calculate the partial derivatives using implicit differentiation.
1. **Step-by-Step Process:**
a. Suppose \(F(T, U, V, W) = (TU - V)^2 \ln(W - UV) - \ln(6) = 0\).
b. We differentiate \(F\) with respect to \(T\) and \(U\) to find the partial derivatives \(\frac{\partial U}{\partial T}\) and \(\frac{\partial T}{\partial U}\).
2. **Apply Implicit Differentiation:**
Let:
\[ F = (TU - V)^2 \ln(W - UV) - \ln(6) \]
Differentiate \(F\) with respect to \(T\) and \(U\):
- To find \(\frac{\partial F}{\partial T}\) and set equal to zero for partials involving \(U\) and \(T\).
- To find \(\frac{\partial F}{\partial U}\) and set equal to zero for partials involving \(T\) and \(U\).
3. **Plug in the values \( (T, U, V, W) = (1, 5, 6, 36) \)**:
Substitute these values in the differentiated equations to solve for \(\frac{\partial U}{\partial T}\) and \(\frac{\partial](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3371c31-de8e-42f8-a5a0-8650bd592ab8%2Fe2731d44-03d2-4141-b99e-12e15ab39571%2Fw0ygaso_processed.png&w=3840&q=75)
Transcribed Image Text:### Implicit Differentiation and Partial Derivatives
#### Problem Statement
Calculate the partial derivatives \( \frac{\partial U}{\partial T} \) and \( \frac{\partial T}{\partial U} \) using implicit differentiation of \((TU - V)^2 \ln(W - UV) = \ln(6)\) at \((T, U, V, W) = (1, 5, 6, 36)\).
(Use symbolic notation and fractions where needed.)
#### Solution
1. **First Partial Derivative \(\frac{\partial U}{\partial T}\)**
\[ \frac{\partial U}{\partial T} = \]
2. **Second Partial Derivative \(\frac{\partial T}{\partial U}\)**
\[ \frac{\partial T}{\partial U} = \]
#### Explanation
Given the equation \((TU - V)^2 \ln(W - UV) = \ln(6)\), we need to calculate the partial derivatives using implicit differentiation.
1. **Step-by-Step Process:**
a. Suppose \(F(T, U, V, W) = (TU - V)^2 \ln(W - UV) - \ln(6) = 0\).
b. We differentiate \(F\) with respect to \(T\) and \(U\) to find the partial derivatives \(\frac{\partial U}{\partial T}\) and \(\frac{\partial T}{\partial U}\).
2. **Apply Implicit Differentiation:**
Let:
\[ F = (TU - V)^2 \ln(W - UV) - \ln(6) \]
Differentiate \(F\) with respect to \(T\) and \(U\):
- To find \(\frac{\partial F}{\partial T}\) and set equal to zero for partials involving \(U\) and \(T\).
- To find \(\frac{\partial F}{\partial U}\) and set equal to zero for partials involving \(T\) and \(U\).
3. **Plug in the values \( (T, U, V, W) = (1, 5, 6, 36) \)**:
Substitute these values in the differentiated equations to solve for \(\frac{\partial U}{\partial T}\) and \(\frac{\partial
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