Obliczyć pochodne cząstkowe drugiego rzędu: Daniel: Obliczyć pochodne cząstkowe drugiego rzędu: f(x,y)= ye^x2y

o nie: w sensie drugie pochodne cząstkowe ?

d2		d2		d2
	i		i		?
dx2		dy2		dxdy

	df		df
licz po kolei pochodne		i		a potem te wyżej
	dx		dy

pochodne po ygreku wymagają wzorku na pochodną iloczynu. Jakiś problem z tą pochodną ?

J: f'_x = ye^x2y2xy = 2y²xe^x2y f'_y = e^x2y + ye^x2yx² = e^x2y(1 + x²y) f'_xx = 2y²(e^x2y + xe^x2y2xy) = 2y²e^x2y(1 + 2x²y) .... dalej licz sam

Dziadek Mróz: f(x, y) = ye^x2y y = f(x, y) y = ye^x2y y = uv u = y v = e^z z = jk j = x² k = y

d		d		d		d
	y =		[uv] = v		u + u		v = (1) ...
dx		dx		dx		dx

d		d
	u =		[y] = 0
dx		dx

d		d		d
	v =		[ez] = ez *		z = (2) ...
dx		dx		dx

d		d		d		d
	z =		[jk] = k		j + j		k = (3) ...
dx		dx		dx		dx

d		d
	j =		[x2] = 2x
dx		dx

d		d
	k =		[y] = 0
dx		dx

... (3) = y2x = 2xy ... (2) = e^x2y * 2xy = 2xye^x2y ... (1) = y2xye^x2y = 2xy²e^x2y

J: Leć dalej... sprawdzę, czy się nie pomyliłem... na razie f'_x mamy tak samo...

Dziadek Mróz: z tych samych zmiennych:

d		d		d		d
	y =		[uv] = v		[u] + u		[v] = (1) ...
dy		dy		dy		dy

d		d
	u =		[y] = 1
dy		dy

d		d		d
	v =		[ez] = ez *		z = (2) ...
dy		dy		dy

d		d		d		d
	z =		[jk] = k		[j] + j		[k] = (3) ...
dy		dy		dy		dy

d		d
	j =		[x2] = 0
dy		dy

d		d
	k =		[y] = 1
dy		dy

... (3) = x² ... (2) = e^x2y * x² = x²e^x2y ... (1) = e^x2y + yx²e^x2y = e^x2y(1 + x²y) =====================================================================

	d
y =		f(x, y)
	dx

y = 2xy²e^x2y y = 2uv u = xy² v = e^z z = x²y

d		d		d		d		d
	y =		[2uv] = 2		[uv] = 2(v		[u] + u		[v]) = (1) ...
dx		dx		dx		dx		dx

d		d		d
	u =		[xy2] = y2		[x] = y2
dx		dx		dx

d		d		d
	v =		[ez] = ez *		[z] = (2) ...
dx		dx		dx

d		d		d
	z =		[x2y] = y		[x2] = 2xy
dx		dx		dx

... (2) = e^x2y * 2xy = 2xye^x2y ... (1) = 2(e^x2yy² + xy²2xye^x2y) = 2(y²e^x2y + 2x²y³e^x2y) = = 2y²e^x2y(1 + 2x²y) −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

d		d		d		d
	y = ... = 2		[uv] = 2(v		[u] + u		[v]) = (1) ...
dy		dy		dy		dy

d		d		d
	u =		[xy2] = x		[y2] = 2xy
dy		dy		dy

d		d		d
	v =		[ez] = ez *		[z] = (2) ...
dy		dy		dy

d		d		d
	z =		[x2y] = x2		[y] = x2
dy		dy		dy

... (2) = e^x2y * x² = x²e^x2y ... (1) = 2(e^x2y2xy + xy²x²e^x2y) = 2(2xye^x2y + x³y²e^x2y) = = 2xye^x2y(2 + x²y) =====================================================================

	d
y =		f(x, y)
	dy

y = e^x2y(1 + x²y) y = uv u = e^z v = 1 + x²y z = x²y

d		d		d		d
	y =		[uv] = v		[u] + u		[v] = (1) ...
dx		dx		dx		dx

d		d		d
	u =		[ez] = ez *		[z] = (2) ...
dx		dx		dx

d		d		d
	v =		[1 + x2y] =		[x2y] = 2xy
dx		dx		dx

d		d
	z =		[x2y] = 2xy
dx		dx

... (2) = e^x2y2xy = 2xye^x2y ... (1) = (1 + x²y)2xye^x2y + e^x2y2xy = 2xye^x2y(2 + x²y) −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

d		d		d		d
	y =		[uv] = v		[u] + u		[v] = (1) ...
dy		dy		dy		dy

d		d		d
	u =		[ez] = ez *		[z] = (2) ...
dy		dy		dy

d		d		d
	v =		[1 + x2y] =		[x2y] = x2
dy		dy		dy

d		d
	z =		[x2y] = x2
dy		dy

... (2) = e^x2yx² = x²e^x2y ... (1) = (1 + x²y)x²e^x2y + e^x2yx² = x²e^x2y(2 + x²y) ===================================================================== ===================================================================== =====================================================================

d
	f(x, y) = 2xy2ex2y
dx

d
	f(x, y) = ex2y(1 + x2y)
dy

d2
	f(x, y) = 2y2ex2y(1 + 2x2y)
dx2

d2
	f(x, y) = 2xyex2y(2 + x2y)
dxdy

d2
	f(x, y) = x2ex2y(2 + x2y)
dy2

d2
	f(x, y) = 2xyex2y(2 + x2y)
dydx

Dziadek Mróz: Pisałem jak wynikało bez sprawdzania