Math Help

Partial Derivatives

Example: For the given function, f(x,y), find f_x(x,y), f_y(x,y), f_xy(x,y), and f_yx(x,y)

f(x,y) = x² exp (- y² )

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

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

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

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

Note: For continuous functions f_xy(x,y) = f_yx(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 = f_xyz

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)

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