Find the first partial derivatives of the function. \(f(x, y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\)

The monthly profit (in dollars) of Bond and Barker Department Store depends on the level of inventory \(x\) (in thousands of dollars) and the floor space \(y\) (in thousands of square feet) available for display of the merchandise, as given by the equation $$\begin{aligned}P(x, y)=&-0.02 x^{2}-15 y^{2}+x y \\ &+39 x+25 y-20,000\end{aligned}$$ Compute \(\partial P / \partial x\) and \(\partial P / \partial y\) when \(x=4000\) and \(y=150\). Interpret your results. Repeat with \(x=5000\) and \(y=150\).

Find the first partial derivatives of the function. \(h(r, s, t)=e^{r s t}\)

Drafted by an international committee in 1989 , the rules for the new International America's Cup Class (IACC) include a formula that governs the basic yacht dimensions. The formula $$f(L, S, D) \leq 42$$ where $$f(L, S, D)=\frac{L+1.25 S^{1 / 2}-9.80 D^{1 / 3}}{0.388}$$ balances the rated length \(L\) (in meters), the rated sail area \(S\) (in square meters), and the displacement \(D\) (in cubic meters). All changes in the basic dimensions are trade-offs. For example, if you want to pick up speed by increasing the sail area, you must pay for it by decreasing the length or increasing the displacement, both of which slow down the boat. Show that yacht A of rated length \(20.95 \mathrm{~m}\), rated sail area $277.3 \mathrm{~m}^{2}\(, and displacement \)17.56 \mathrm{~m}^{3}$ and the longer and heavier yacht \(\mathrm{B}\) with \(L=21.87, S=311.78\), and \(D=\) \(22.48\) both satisfy the formula.

Find the first partial derivatives of the function. \(g(x, y)=2 x^{2}+4 y+1\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \((a, b)\) is a critical point of \(f\) and both the conditions $f_{x x}(a, b)<0\( and \)f_{y y}(a, b)<0\( hold, then \)f\( has a relative maximum at \)(a, b)$.