91Ó°ÊÓ

First, we need to find the partial derivatives of Q with respect to r and s. $$\frac{\partial Q}{\partial r} = \frac{1}{s}$$ $$\frac{\partial Q}{\partial s} = -\frac{r}{s^2}$$

Now, let's find the second order partial derivative with respect to r by differentiating the first order partial derivative with respect to r itself. $$\frac{\partial^2Q}{\partial r^2} = \frac{\partial}{\partial r}\left(\frac{\partial Q}{\partial r}\right) = \frac{\partial}{\partial r}\left(\frac{1}{s}\right) = 0$$

Next, we will find the second order partial derivative with respect to s by differentiating the first order partial derivative with respect to s itself. $$\frac{\partial^2Q}{\partial s^2} = \frac{\partial}{\partial s}\left(\frac{\partial Q}{\partial s}\right) = \frac{\partial}{\partial s}\left(-\frac{r}{s^2}\right) = \frac{2r}{s^3}$$

Finally, we will find the mixed second order partial derivatives by differentiating the first order partial derivatives with respect to the other variable. $$\frac{\partial^2Q}{\partial r \partial s} = \frac{\partial}{\partial r}\left(\frac{\partial Q}{\partial s}\right) = \frac{\partial}{\partial r}\left(-\frac{r}{s^2}\right) = -\frac{1}{s^2}$$ $$\frac{\partial^2Q}{\partial s \partial r} = \frac{\partial}{\partial s}\left(\frac{\partial Q}{\partial r}\right) = \frac{\partial}{\partial s}\left(\frac{1}{s}\right) = -\frac{1}{s^2}$$ In summary, the four second partial derivatives are as follows: 1. $$\frac{\partial^2Q}{\partial r^2} = 0$$ 2. $$\frac{\partial^2Q}{\partial s^2} = \frac{2r}{s^3}$$ 3. $$\frac{\partial^2Q}{\partial r \partial s} = -\frac{1}{s^2}$$ 4. $$\frac{\partial^2Q}{\partial s \partial r} = -\frac{1}{s^2}$$