Problem 46 - Calculus Early Transcendentals

Question

Find the first partial derivatives of the following functions. $$G(r, s, t)=\sqrt{r s+r t+s t}$$

Step-by-Step Solution

Answer

Answer: The first partial derivatives of the function G(r, s, t) are: $$\frac{\partial G}{\partial r} = \frac{s + t}{2\sqrt{r s+r t+s t}}$$ $$\frac{\partial G}{\partial s} = \frac{r + t}{2\sqrt{r s+r t+s t}}$$ $$\frac{\partial G}{\partial t} = \frac{r + s}{2\sqrt{r s+r t+s t}}$$

Step 1: Rewrite the function in a more convenient form

Let's rewrite G(r, s, t) as: $$G(r, s, t) = (r s+r t+s t)^{1/2}$$

Step 2: Determine the partial derivative with respect to r

Apply the chain rule, and differentiate G(r, s, t) with respect to r: $$\frac{\partial G}{\partial r} = \frac{1}{2}(r s+r t+s t)^{-1/2} \cdot \frac{\partial}{\partial r}(r s+r t+s t)$$ Now differentiate the inside of the parentheses with respect to r: $$\frac{\partial}{\partial r}(r s+r t+s t) = s + t$$ Plug this back into the derivative: $$\frac{\partial G}{\partial r} = \frac{1}{2}(r s+r t+s t)^{-1/2} \cdot (s + t)$$

Step 3: Determine the partial derivative with respect to s

Following the same procedure, differentiate G(r, s, t) with respect to s: $$\frac{\partial G}{\partial s} = \frac{1}{2}(r s+r t+s t)^{-1/2} \cdot \frac{\partial}{\partial s}(r s+r t+s t)$$ Now differentiate the inside of the parentheses with respect to s: $$\frac{\partial}{\partial s}(r s+r t+s t) = r + t$$ Plug this back into the derivative: $$\frac{\partial G}{\partial s} = \frac{1}{2}(r s+r t+s t)^{-1/2} \cdot (r + t)$$

Step 4: Determine the partial derivative with respect to t

Lastly, differentiate G(r, s, t) with respect to t: $$\frac{\partial G}{\partial t} = \frac{1}{2}(r s+r t+s t)^{-1/2} \cdot \frac{\partial}{\partial t}(r s+r t+s t)$$ Now differentiate the inside of the parentheses with respect to t: $$\frac{\partial}{\partial t}(r s+r t+s t) = r+ s$$ Plug this back into the derivative: $$\frac{\partial G}{\partial t} = \frac{1}{2}(r s+r t+s t)^{-1/2} \cdot (r + s)$$

Step 5: Give final results for the partial derivatives

The first partial derivatives of the function G(r, s, t) are: $$\frac{\partial G}{\partial r} = \frac{s + t}{2\sqrt{r s+r t+s t}}$$ $$\frac{\partial G}{\partial s} = \frac{r + t}{2\sqrt{r s+r t+s t}}$$ $$\frac{\partial G}{\partial t} = \frac{r + s}{2\sqrt{r s+r t+s t}}$$