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.