这是一份bath巴斯大学PH10007/PH20107作业代写的成功案

We can see immediately that
$$
\frac{\partial f}{\partial x}=2 x+3 y, \quad \frac{\partial f}{\partial y}=3 x, \quad \frac{d y}{d x}=\frac{1}{\left(1-x^{2}\right)^{1 / 2}}
$$
and so, using (5.8) with $x_{1}=x$ and $x_{2}=y$,
$$
\begin{aligned}
\frac{d f}{d x} &=2 x+3 y+3 x \frac{1}{\left(1-x^{2}\right)^{1 / 2}} \
&=2 x+3 \sin ^{-1} x+\frac{3 x}{\left(1-x^{2}\right)^{1 / 2}}
\end{aligned}
$$
Obwiously the same expression would have resulted if we had substituted for $y$ from the start, but the above method often produces results with reduced calculation, particularly in more complicated examples.

PH20107 COURSE NOTES ：

In the above notation, $A(x, y)=3 y$ and $B(x, y)=x$ and so
$$
\frac{\partial A}{\partial y}=3, \quad \frac{\partial B}{\partial x}=1 .
$$
As these are not equal it follows that the differential is inexact.
Determining whether a differential containing many variable $x_{1}, x_{2}, \ldots, x_{n}$ is exact is a simple extension of the above. A differential containing many variables can be written in general as
$$
d f=\sum_{i=1}^{n} g_{i}\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{i}
$$
and will be exact if
$$
\frac{\partial g_{i}}{\partial x_{j}}=\frac{\partial g_{j}}{\partial x_{i}} \quad \text { for all pairs } i, j
$$
There will be $\frac{1}{2} n(n-1)$ such relationships to be satisfied.