二変数関数 $f(x, y)$ の二階偏導関数¶
Step 1: 二変数関数の微分¶
$$ \begin{align*} &f(x,y) \equiv \frac{n!}{a! b! (n-a-b)!} x^a (y-x)^b (1-y)^{n-a-b}\\ &\text{want to show}\Rightarrow\frac{\partial f(x, y)}{\partial x\partial y} \end{align*} $$In [2]:
f[x_, y_] := (n!/(a! * b! * (n-a-b)!)) * x^a * (y-x)^b * (1-y)^{n-a-b}
D[f[x, y],x, y]
Out[2]:
In [3]:
ToString[TeXForm[D[f[x, y],x, y]]]
Out[3]:
\left\{\frac{a b n! x^{a-1} (y-x)^{b-1} (1-y)^{-a-b+n}}{a! b! (-a-b+n)!}-\frac{a n!\
> x^{a-1} (-a-b+n) (y-x)^b (1-y)^{-a-b+n-1}}{a! b! (-a-b+n)!}-\frac{(b-1) b n! x^a\
> (y-x)^{b-2} (1-y)^{-a-b+n}}{a! b! (-a-b+n)!}+\frac{b n! x^a (-a-b+n) (y-x)^{b-1}\
> (1-y)^{-a-b+n-1}}{a! b! (-a-b+n)!}\right\}In [4]:
CopyToClipboard[ToString[TeXForm[D[f[x, y],x, y]]]]