Given that z is a smooth function of x and y and zxy = -4x, we can use the symmetry of mixed partial derivatives to determine the values of the following:
1. zyx: By the symmetry of mixed partial derivatives, zyx will be equal to -4y. So, zyx = -4y.
2. zxyx: This is the second derivative of z with respect to x and y, followed by the first derivative with respect to x. Since z is a smooth function, the order of differentiation does not matter. So, zxyx will be equal to the second derivative of z with respect to x and y, which can be represented as z_{xy} = -4x. Therefore, zxyx = -4x.
3. zxyy: This is the second derivative of z with respect to x and y, followed by the first derivative with respect to y. Since z is a smooth function, the order of differentiation does not matter. So, zxyy will be equal to the second derivative of z with respect to x and y, which can be represented as z_{xy} = -4x. Therefore, zxyy = -4x.
Assume z
is a smooth function of x
and y
. If zxy=−4x
, what can you say about each of the following?
zyx=
zxyx=
zxyy=
1 answer