Question

Seja f(x,y) =$x\:sen( 3x+2y^2)$ . Mostre que $\dfrac{\Delta^2 f}{\Delta x\Delta y} =\dfrac{\Delta^2 f}{\Delta y\Delta x}$

Answer 1

Resposta:

f(x,y) = x * sen(3x+2y²)

df/dx=fx

fx= 1 * sen(3x+2y²) + x * [sen(3x+2y²)]'

fx= sen(3x+2y²) + x * cos(3x+2y²)  * (3x+2y²)'

fx= sen(3x+2y²) + x * cos(3x+2y²)  * 3

fx= sen(3x+2y²) + 3*x * cos(3x+2y²)

fxy=cos(3x+2y²)*4y -3x sen(3x+2y²) * 4y

fxy=cos(3x+2y²)*4y -12xy sen(3x+2y²)

fxy=4y*cos(3x+2y²) -12xy sen(3x+2y²)

fy=x*cos(3x+2y²)*(3x+2y²)'

fy=x*cos(3x+2y²)*4y

fy=4yx*cos(3x+2y²)

fyx= 4y*cos(3x+2y²) -4yx *sen(3x+2y²)* (3x+2y²)'

fyx= 4y*cos(3x+2y²) -4yx *sen(3x+2y²)* 3

fyx= 4y*cos(3x+2y²) -12yx *sen(3x+2y²)

Como podemos ver fxy=fyx ........................c.q.p.

** c.q.p. como queríamos provar