Question

if f=(2x²y-x⁴)î+(e^xy-y sin x)j cap x²cosy k cap verify that d²f/dxdy = d²f/dydx​

Answer 1

If f(x, y) = 8 x3y2 + 3 x y-2 then

df/dx = 24 x2 y2 + 3 y-2 and df/dy = 16 x3y - 6 x y-3

There are three second partial derivatives, written

d2f(x, y)/dx2, d2f(x, y)/dxdy and d2f(x, y)/dy2

which mean, respectively, the second derivative of f(x, y) with respect to x with y constant; the derivative w.r.t. x of the derivative w.r.t. y of f(x, y); and the second derivative of f(x, y) w.r.t y with x held constant.

Answer 2

If f(x, y) = x2y + x sin y, then

df/dx = 2 x y + sin y and df/dy =x2 + x cos y.

Example II If f(x, y) = 8 x3y2 + 3 x y-2 then

df/dx = 24 x2 y2 + 3 y-2 and df/dy = 16 x3y - 6 x y-3

There are three second partial derivatives, written

d2f(x, y)/dx2, d2f(x, y)/dxdy and d2f(x, y)/dy2

which mean, respectively, the second derivative of f(x, y) with respect to x with y constant; the derivative w.r.t. x of the derivative w.r.t. y of f(x, y); and the second derivative of f(x, y) w.r.t y with x held constant.

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