Parcijalna derivacija drugog reda

Odredimo najprije parcijalne derivacije prvog reda zadane funkcije. Vrijedi

$\displaystyle \frac{\partial f}{\partial x}\left( x,y\right)=yx^{y-1}$ i $\displaystyle \frac{\partial f}{\partial y}\left( x,y\right)=x^y\ln x.$

Sada je prema

[M2, definicija 3.8]

$\displaystyle \frac{\partial ^2 f}{\partial x^2}\left( x,y\right)$	$\displaystyle = \frac{\partial }{\partial x}\left(yx^{y-1}\right)=y(y-1)x^{y-2},$
$\displaystyle \frac{\partial ^2 f}{\partial x\partial y}\left( x,y\right)$	$\displaystyle = \frac{\partial }{\partial x}\left(x^y\ln x\right)=yx^{y-1}\ln x+x^{y-1},$
$\displaystyle \frac{\partial ^2f}{\partial y^2}\left( x,y\right)$	$\displaystyle =\frac{\partial }{\partial y}\left(x^y\ln x\right)=x^y(\ln x)^2.$

Parcijalne derivacije prvog reda funkcije

u proizvoljnoj točki $\displaystyle \left( x,y\right)\in\mathcal{D}_f$ glase:

$\displaystyle \frac{\partial f}{\partial x}\left( x,y\right)$	$\displaystyle = m(1+x)^{m-1}(1+y)^n,$
$\displaystyle \frac{\partial f}{\partial y}\left( x,y\right)$	$\displaystyle = n(1+x)^m(1+y)^{n-1}.$

Parcijalne derivacije drugog reda funkcije

u proizvoljnoj točki $\displaystyle \left( x,y\right)\in\mathcal{D}_f$ su:

$\displaystyle \frac{\partial ^2 f}{\partial x^2}\left( x,y\right)$	$\displaystyle = m(m-1)(1+x)^{m-2}(1+y)^n,$
$\displaystyle \frac{\partial ^2 f}{\partial x\partial y}\left( x,y\right)$	$\displaystyle = mn(1+x)^{m-1}(1+y)^{n-1},$
$\displaystyle \frac{\partial ^2f}{\partial y^2}\left( x,y\right)$	$\displaystyle = n(n-1)(1+x)^m(1+y)^{n-2}.$

Prema tome, vrijednosti parcijalnih derivacija drugog reda funkcije

u točki

su:

$\displaystyle \frac{\partial ^2 f}{\partial x^2}(0,0)=m(m-1), \quad \frac{\pa... ...ial x\partial y}(0,0)=mn, \quad \frac{\partial ^2f}{\partial y^2}(0,0)=n(n-1).$