First, the partial derivative of multivariate composite function

The above formula can be simply written as "line multiplication, line addition"; We can also use the invariance of differential form, that is, if the function has several intermediate variables, the partial derivative consists of several parts (not excluding some parts being zero).

Second, the second partial derivative of multivariate composite function

For the second-order partial derivative of the composite function, it needs to be understood that the partial derivative of the function to the intermediate variable is still a multivariate composite function, and its relationship is completely consistent with the original relationship between the dependent variable and the independent variable, namely:

Draw a picture first:

The key to solve the problem of higher-order partial derivatives of multivariate compound abstract functions is to clarify the relationship between dependent variables and independent variables, and finally draw a relationship diagram in the process of solving problems to avoid overwriting or omission.

：

Geometric meaning of partial derivative;

Represents the tangent slope of a point on a fixed surface.

The partial derivative f'x(x0, y0) represents the tangent slope of a point on a fixed surface to the X axis; The partial derivative f'y(x0, y0) represents the tangent slope of a point on a fixed surface with respect to the y axis.

Higher-order partial derivatives: If the partial derivatives f'x(x, y) and f'y(x, y) of the binary function z=f(x, y) are still derivable, then the partial derivatives of these two partial derivatives are called the second-order partial derivatives of z=f(x, y). Binary function has four second-order partial derivatives: f"xx, f"xy, f"yx, f"yy.

The difference between f "xy" and f "yx" is that the former takes the partial derivative of x first, and then takes the partial derivative of y from the obtained partial derivative function; The latter takes the partial derivative of y first, and then the partial derivative of x. When f "xy" and f "yx" are continuous, the result of derivative is independent of order.