Find the first and second derivatives of the

Use polar coordinates to find the limit. [Hint: Let ${\displaystyle x=r\cos \text{and}y=r\sin }$ , and note that (x, y) (0, 0) implies r 0.] ${\displaystyle \underset{(x,y)\to (0,0)}{lim}\frac{x^2-y^2}{\sqrt{x^2+y^2}}}$

Lynbrook West , an apartment complex , has 100 two-bedroom units. The montly profit (in dollars) realized from renting out x apartments is given by the following function.
${\displaystyle P\left(x\right)=-12x^2+2136x-41000}$
To maximize the monthly rental profit , how many units should be rented out?
What is the maximum monthly profit realizable?

Differentiation of multivariable function proof
${\displaystyle \frac{d}{dx}{\int }_{v\left(x\right)}^{u\left(x\right)}f(t,x)dt=u^{\prime }\left(x\right)f(u\left(x\right),x)-v^{\prime }\left(x\right)f(v\left(x\right),x)+{\int }_{v\left(x\right)}^{u\left(x\right)}\frac{\partial }{\partial x}f(t,x)dt}$

Let f1,…,fr be complex polynomials in the variables x1,…,xn let V be the variety of their common zeros, and let I be the ideal of the polynomial ring $R=C[x1,\dots ,xn]$ that they generate. Define a homomorphism from the quotient ring $\bar{R}=Rx2F$; I to the ring RR of continuous, complex-valued functions on V.

In a university examination, which was indeed very tough, ${\displaystyle 50\mathrm{\%}}$ at jeast failed in "Statistics",${\displaystyle 75\mathrm{\%}}$ at least in Topology, ${\displaystyle 82\mathrm{\%}}$ at least in "Functional Analysis" and ${\displaystyle 96\mathrm{\%}}$ at least in "Applied Mathematics". How many at least failed in all the four? (Ans.${\displaystyle 3\mathrm{\%}}$)

A quick question; is it possible to say in a way analogous to the single variable case that a multivariable function is "asymptotically equivalent" to a second multivariable function? For example, consider the function of ${\displaystyle n_1,n_2\in \mathbb{R}}$ given by
${\displaystyle \text{Var}\left(\hat{\mu }\right)=\frac{{\sigma }^2(n_1+2n_2)}{{(n_1+n_2)}^2}.}$
where ${\displaystyle {\sigma }^2}$ is a constant.
Can we say that ${\displaystyle \text{Var}\left(\hat{\mu }\right)\approx \frac{1}{n_1+n_2}}$ and then conclude that ${\displaystyle \text{Var}\left(\hat{\mu }\right)\to 0}$ as ${\displaystyle n_1\to \mathrm{\infty }}$ and ${\displaystyle n_2\to \mathrm{\infty }}$?
${\displaystyle \underset{(x,y)\to (\mathrm{\infty },\mathrm{\infty })}{lim}\frac{x+2y}{{(x+y)}^2}}$
does not exist. Am I wrong to think of ${\displaystyle \text{Var}\left(\hat{\mu }\right)}$ as a function of two variables?

Use the Chain Rule to find ${\displaystyle \frac{dw}{dt}}$, where ${\displaystyle w=\sin 8x\cos 2y,x=\frac{t}{2}\text{ and }y=t^4}$
${\displaystyle \frac{dw}{dx}}$
(Type an expression using x and y as the variables)
${\displaystyle \frac{dw}{dy}}$
(Type an expression using x and y as the variables)
${\displaystyle \frac{dy}{dt}}$
(Type an expression using t as the variable )
${\displaystyle \frac{dw}{dt}}$
(Type an expression using t as the variable )