Derivative emphasizes the slope of the curve and the rate of change of variables;
What can be emphasized a little is separability, continuity and smoothness.
Dx, dy: differentiability; Dy/dx: differentiability
Dy = (dy/dx)dx, which becomes: Δ y = (dy/dx) Δ x in engineering application.
This is the relationship between differentiability and differentiability:
Differentiable = differentiable.
Derivative = differential = differential, derivative
Non-derivative = non-differentiable = undefined
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2. Bivariate functions and multivariate functions with more than two variables have the concept of partial derivatives.
There are concepts of total differential and total differential.
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Multivariate function has the concept of directional differential/derivative.
The unary function has no partial derivative, total derivative, total differential, partial derivative and directional derivative.
3. For multivariate functions, the derivatives along any coordinate axis are partial derivatives.
A derivative in any particular direction is a directional derivative.
B. the directional derivative with the maximum value is the gradient.
C, there is the concept of total difference in English, but we are not used to teaching.
So called, we are used to calling it total differential, but it actually means complete equivalence.
Unary functions do not have these concepts. Partial derivative is full derivative, and full derivative is partial derivative.
4, dx, dy and du are all differential, and only written as du= (? f/？ x)dx +(？ f/？ Y) death,
Du is fully differential, dx and dy are partial differential, but we are not used to saying so.
And then what? F, is it? X, right? Y is also the concept of differentiation, which is the deformation of df, dx and dy in multivariate functions.
A single change in x will cause a change in u, du= (? f/？ x)dx
A single change in y will cause a change in u, du= (? f/？ y)dy
One of them? f/？ X, right? f/？ Y is the partial derivative of binary function f to x and y respectively.
f/？ X is the change rate of F caused by the change of X alone, which is caused by some reasons and is "partial";
f/？ Y is the change rate of F caused by the change of Y alone, which is caused by some reasons and is "partial".
X, y change at the same time, cause the change of u is:
Du = (? f/？ x)dx +(？ f/？ y)dy
This is the total differential, and all the causes of * * * are caused by "all".
To sum up, in a word:
For unary functions, there is no essential difference between derivable and derivable;
For multivariate functions, differentiability means that all directions can be differentiated, which requires higher differentiability.