total differential s=p/q with step by step solution

To find the total differential of the function \( s = \frac{p}{q} \), let's go through the process step by step:

### 1. **Identify the variables:**
   - We have a function \( s = \frac{p}{q} \), where \( p \) and \( q \) are both functions of some independent variables, say \( x \) and \( y \). We want to compute the total differential \( ds \).

### 2. **Formula for the total differential:**
   The total differential \( ds \) is given by:
   \[
   ds = \frac{\partial s}{\partial p} \, dp + \frac{\partial s}{\partial q} \, dq
   \]
   Here, we need to find the partial derivatives of \( s \) with respect to both \( p \) and \( q \).

### 3. **Compute the partial derivatives:**
   
   - The partial derivative of \( s \) with respect to \( p \):
     \[
     \frac{\partial s}{\partial p} = \frac{1}{q}
     \]
   
   - The partial derivative of \( s \) with respect to \( q \):
     \[
     \frac{\partial s}{\partial q} = -\frac{p}{q^2}
     \]

### 4. **Substitute the partial derivatives into the total differential formula:**
   Using the results from the partial derivatives, we now substitute them back into the total differential formula:
   \[
   ds = \frac{1}{q} \, dp + \left(-\frac{p}{q^2}\right) \, dq
   \]

### 5. **Final expression for the total differential:**
   Therefore, the total differential of \( s = \frac{p}{q} \) is:
   \[
   ds = \frac{1}{q} \, dp - \frac{p}{q^2} \, dq
   \]

This expression represents the total differential of \( s \) in terms of the differentials \( dp \) and \( dq \).

Would you like to discuss this solution in more detail or ask any questions? Here are some related questions to explore further:

1. How does the total differential relate to the chain rule in calculus?
2. What happens to the total differential if \( p \) and \( q \) are independent of each other?
3. How can we use the total differential to approximate changes in \( s \)?
4. Can you explain the geometric interpretation of the total differential?
5. How do partial derivatives factor into the computation of total differentials for multivariable functions?

**Tip:** The total differential provides a linear approximation of a function's change based on the changes in its variables, which is especially useful in applied mathematics.

This detailed step-by-step solution explains how to find the total differential of s = p/q, utilizing partial derivatives. Learn about the calculus concept of total differentials, including the key formulas involved. Ideal for students in Grades 11-12 or early college.

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