The derivative obtained by taking the derivative in a certain direction at the inner point of the function definition domain. Generally, it is the directional derivative of binary function and ternary function, and the directional derivative can be divided into the directional derivative along a straight line and the directional derivative along a curve.

In mathematics, the partial derivative of a multivariable function is its derivative of one variable, while keeping other variables unchanged (compared with the total derivative, all variables in the total derivative are allowed to change). Partial derivatives are very useful in vector analysis and differential geometry.

The directional derivative is expressed by partial derivative.

The popular explanation of directional derivative is not only to know the rate of change of function in the coordinate axis direction (that is, partial derivative), but also to try to find out the rate of change of function in other specific directions. The directional derivative is the rate of change of a function in other specific directions.

In mathematics, the partial derivative of a multivariable function is its derivative of one variable, while keeping other variables unchanged (compared with the total derivative, all variables in the total derivative are allowed to change).

Can the existence of directional derivatives in any direction be inferred from the existence of partial derivatives? -No..

It can only be inferred that the directional derivative along each coordinate axis (such as X axis) exists, but if the directional derivative along the positive semi-axis direction of X axis and the directional derivative along the negative semi-axis direction of X axis are not opposite, then the partial derivative about X does not exist.

This is similar to the existence of the left and right derivatives of a univariate function at a certain point, which does not mean that the point derivative exists.