Teacher, I would like to ask you about the slope of derivative. The minimum tangent of slope is parallel to 12x + y = 6. Then the calculation process of the slope of the tangent is detailed

Teacher, I would like to ask you about the slope of derivative. The minimum tangent of slope is parallel to 12x + y = 6. Then the calculation process of the slope of the tangent is detailed

y=-12x+6
Parallel to this line, the slope is equal to - 12

Derivative and differential, given f (x) and the tangent slope k of a point, how to find the normal equation of the point, list the formula

y-y0=-1/k(x-x0)

Sufficient conditions for differentiability of binary functions
A sufficient condition for the differentiation of binary functions is that the partial derivatives of X and Y exist and are continuous
Isn't differentiability differentiable for any direction? If only two partial derivatives, differentiability can be deduced?

It is true that there are strict proofs in this book. Mathematical research relies on proofs obtained from definitions and theorems. Although some facts are intuitively not easy to understand, they should be admitted after proofs

Sufficient conditions for differentiability of binary functions
Sufficient conditions for differentiability of binary function f (x, y) at point (0,0)
Besides the existence of partial derivatives, what conditions should be satisfied?

Two partial derivatives exist and are continuous at (0,0)
Reminder: if the partial derivative is discontinuous, the function may also be differentiable

Sufficient conditions for differentiability of binary functions at a certain point

The sufficient condition is that the two partial derivatives at the point are continuous, and the necessary condition is that the two partial derivatives at the point exist

In bivariate function, why continuity is not necessarily differentiable and partial derivative exists

Function of one variable is not necessarily differentiable and differentiable, let alone function of two variables

Please explain in detail the concepts and relations of differentiability, differentiability and continuity of functions of one variable and two variables,

Unary: differentiable is equivalent to differentiable, differentiable can deduce continuous, continuous can not deduce differentiable
Binary: the partial derivative can be derived continuously from differentiable, differentiable from continuous, differentiable from existence of partial derivative

How to prove whether the function of two variables is differentiable? What is the specific method? Urgent, better detailed@

Hello: Thank you for asking our team for help! If binary, we need to verify whether AZ / P is 0 (AZ is the high order infinitesimal of Z), if it is 0, it is differentiable, otherwise it is not differentiable. (where p = root [(AX) ^ 2 + (Ay) ^ 2] (ax, ay is the high order infinitesimal of X, y)) some symbols are well typed, this is a method to verify differentiability, AZ needs

How to prove the differentiability of this high number binary function?
 

What is the geometric meaning of continuous, partial derivative, but nondifferentiable functions of variables?
Continuous means continuous images,
A partial derivative is smooth in one direction,
What is the geometric meaning of differentiability?

If y = f (x) is differentiable, then dy = f '(x) DX, and Dy and DX are understood as tiny increments of Y and X at x0, that is dy = y-y0, DX = x-x0, then the differentiable expression becomes y-y0 = f' (x0) (x-x0), which is the tangent equation of F (x) image at x0, and differentiability means the existence of tangent equation, y) The total differential expression of DZ = z'x * DX + z'y * dy is actually the tangent plane equation of binary function at (x0, Y0). Therefore, if the binary function is not differentiable at a certain point, it means that there is no tangent plane in the function image at that point