Partial
Derivatives
Example: 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. |
Copyright © 2017 Richard C. Coddington
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