Partial derivative: the rate of change of the function value caused by the change of the independent variable when the function delays the positive direction of the coordinate axis at a certain point.

Directional derivative: the rate of change of function value caused by the change of independent variable in any direction at a certain point.

Therefore, their differences are mainly as follows:

1, obviously, the partial derivative is only the direction of the coordinate axis, and the directional derivative is arbitrary;

2. So when we find the directional derivative along the coordinate axis, will the result be the same as the partial derivative? We see that if we find the directional derivative along the positive axis, it is the same as the partial derivative; If the directional derivative is found in the direction of "negative direction along the coordinate axis", the result differs from the partial derivative by a negative sign.

Reciprocal direction is equivalent to vector class. If y = the absolute value of x, the directional derivative at o exists, the left directional derivative is-1, and the right directional derivative is 1, but the partial derivative at 0 does not exist. In space, if the partial derivative exists, then the tangent of that point in that direction exists, but the directional derivative exists, which only shows that the ray exists. Similar to the relationship between the left limit and the right limit of a point.