Problem 33 - Calculus Early Transcendentals

Question

Find the four second partial derivatives of the following functions. $$F(r, s)=r e^{s}$$

Step-by-Step Solution

Answer

Answer: The four second partial derivatives are: 1. $$\frac{\partial^2 F}{\partial r^2} = 0$$ 2. $$\frac{\partial^2 F}{\partial s^2} = r e^s$$ 3. $$\frac{\partial^2 F}{\partial r \partial s} = e^s$$ 4. $$\frac{\partial^2 F}{\partial s \partial r} = e^s$$

Step 1: Find the first partial derivatives with respect to r and s

To find the first partial derivatives, we will treat one variable as constant while differentiating with respect to the other variable. For the partial derivative with respect to r: $$\frac{\partial F}{\partial r}= \frac{\partial (r e^s)}{\partial r} = e^s$$ For the partial derivative with respect to s: $$\frac{\partial F}{\partial s}= \frac{\partial (r e^s)}{\partial s} = r e^s$$

Step 2: Find the second partial derivatives

Now that we have the first partial derivatives, we'll differentiate them again with respect to r and s. We have four possible combinations of the second partial derivatives: 1. $$\frac{\partial^2 F}{\partial r^2}$$: Differentiating $$\frac{\partial F}{\partial r}= e^s$$ with respect to r: $$\frac{\partial^2 F}{\partial r^2} = \frac{\partial (e^s)}{\partial r} = 0$$ 2. $$\frac{\partial^2 F}{\partial s^2}$$: Differentiating $$\frac{\partial F}{\partial s}= r e^s$$ with respect to s: $$\frac{\partial^2 F}{\partial s^2} = \frac{\partial (r e^s)}{\partial s} = r e^s$$ 3. $$\frac{\partial^2 F}{\partial r \partial s}$$: Differentiating $$\frac{\partial F}{\partial r}= e^s$$ with respect to s: $$\frac{\partial^2 F}{\partial r \partial s} = \frac{\partial (e^s)}{\partial s} = e^s$$ 4. $$\frac{\partial^2 F}{\partial s \partial r}$$: Differentiating $$\frac{\partial F}{\partial s}= r e^s$$ with respect to r: $$\frac{\partial^2 F}{\partial s \partial r} = \frac{\partial (r e^s)}{\partial r} = e^s$$ The four second partial derivatives of the function $$F(r, s)=r e^{s}$$ are: 1. $$\frac{\partial^2 F}{\partial r^2} = 0$$ 2. $$\frac{\partial^2 F}{\partial s^2} = r e^s$$ 3. $$\frac{\partial^2 F}{\partial r \partial s} = e^s$$ 4. $$\frac{\partial^2 F}{\partial s \partial r} = e^s$$