Question

Find f(x,y), f(x,y), f.(1.3), and f,(-2,4) for the given function. If

z=f(x,y)= 3x³y²-x²y³ + 4x +9

Answer 1

Step-by-step explanation:

For the given function, f(x,y), find fx(x,y), fy(x,y), fxy(x,y), and fyx(x,y)

f(x,y) = x2 exp (- y2 )

∂f/∂x = fx(x,y) = 2 x exp(-y2 ) (holding y constant)

∂f/∂y = fy(x,y) = x2 (-2y) exp (- y2 ) (holding x constant)

∂/∂x[∂f/∂y] = fxy(x,y) = 2 x exp(-y2 ) [-2y] = -4 x y exp (-y2 )

∂/∂y[∂f/∂x] = fyx(x,y) = (2x) [-2 y exp(-y2 )] = -4 x y exp (-y2 )

Note: For continuous functions fxy(x,y) = fyx(x,y)

i. e. The order of differentiation is irrelevant.

Example: The concept of partial derivatives can be extended to a function of more than

two independent variables as shown in this example.

Given: f(x, y, z) = cos (4x + 3y + 2z) Find: ∂3f/∂x∂y∂z = fxyz

Start with ∂f/∂x = - 4 sin(4x + 3y + 2z)

Then take the next derivative with respect to y giving ∂2f/∂x∂y = -12 cos(4x + 3y + 2z)

Finally take the derivative with respect to z gives ∂3f/∂x∂y∂z

∂3f/∂x∂y∂z = 24 sin(4x + 3y + 2z) (result)

Note that the cosine function is continuous. So the order of differentiation should be

immaterial. Let us check this out by calculating ∂3f/∂z∂y∂x

Start with ∂f/∂z = - 2 sin(4x + 3y + 2z)

Then ∂2f/∂z∂y = -6 cos(4x + 3y + 2z)

And finally ∂3f/∂x∂y∂z = 24 sin(4x + 3y + 2z) (same result)

Click here to continue with discussion on physical interpretation of the partial derivative.