Problem 32 - Calculus Early Transcendentals

Question

Find the four second partial derivatives of the following functions. $$Q(r, s)=r / s$$

Step-by-Step Solution

Answer

Question: Find the four second partial derivatives of the function Q = ln(r/s). Answer: The four second partial derivatives of the function 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. $$\frac{\partial^2Q}{\partial s \partial r} = -\frac{1}{s^2}$$

Step 1: Find the First Order Partial Derivatives

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

Step 2: Find the Second Order Partial Derivative with respect to r

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. $$\frac{\partial^2Q}{\partial r^2} = \frac{\partial}{\partial r}\left(\frac{\partial Q}{\partial r}\right) = \frac{\partial}{\partial r}\left(\frac{1}{s}\right) = 0$$

Step 3: Find the Second Order Partial Derivative with respect to s

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. $$\frac{\partial^2Q}{\partial s^2} = \frac{\partial}{\partial s}\left(\frac{\partial Q}{\partial s}\right) = \frac{\partial}{\partial s}\left(-\frac{r}{s^2}\right) = \frac{2r}{s^3}$$

Step 4: Find the Mixed Second Order Partial Derivatives

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. $$\frac{\partial^2Q}{\partial s \partial r} = -\frac{1}{s^2}$$