The difference between differentiable and derivable Let s take an example

The only difference between differentiable and conductive:

In the unary function, the derivable and the differentiable are equivalent, they have nothing to do with the integrable, and the multivariate function can be differentiable and derivable, and the vice versa does not hold.

For example, let y=f(x) be a univariate function, if y has a derivative y at x=x[0].'=f'(x), then y is said to be derivable at x=x[0].

If a function is derivable at x[0], then it must be continuous at x[0].

If a function is continuous at x[0], then it is not necessarily derivable at x[0].

Derivative Definition of Function:

1. If f(x) is continuous at x0, then when a tends to 0, [f(x+a)-f(x)] a has a limit, then f(x) is said to be derivable at x0.

2. If f(m) is derivable at any point in the interval (a, b), then f(x) is said to be derivable on (a, b).

The conductible and differential shut-off wisdom system must be differentiable, the file speed must also be directable, and the differentiability and the guidetable are mutually sufficient conditions.

There is a definition in a certain field where the differentiability is located, and when an increment is given, there is also an increment accordingly, and if it can be expressed as, then it is said to be differentiable everywhere.

If the conductive limit exists, it can be derived, and if the limit does not exist, it is not conductive. The same is true for other representations of derivative definitions, which are essentially limits to exist.

Definition: Let the function be defined in the immediate neighborhood, and if , then it is said to be continuous at the point. Theorem:

If and only then, existed. That is, the left limit and the right limit exist and are equal, and the limit exists. The conditions that are continuously required to be met are:

to have a definition in a certain neighborhood; Limits exist.

Relation between derivable and differentiable: differentiable = > derivable = > continuous = > integrable, in a monary function, derivable and differentiable are equivalent.

The relationship between derivable and continuous: continuous must be continuous, and continuous is not necessarily derivable.

Differentiable vs. Continuous: Differentiable and Derivable are the same.

The relationship between integrable and continuous: integrable is not necessarily continuous, continuous must be integrable.

The relationship between derivable and integrable: Derivable is generally integrable, and integrable can not be deduced from a certain derivative.

Differentiable = > derivable = > continuous = > integrable.

Differentiable conditions

Prerequisite. If a function is differentiable at a point, the function must be continuous at that point.

If a binary function is differentiable at a point, then the partial derivatives of the function for x and y must exist at that point.

Sufficient condition. If the partial derivatives of the function for x and y exist in a neighborhood at this point, and are continuous at this point, then the function is differentiable at this point.

Guideable conditions. Sufficient and necessary conditions: sufficient and necessary conditions for the function to be derivable: the function is continuous at that point and the left and right derivatives are both blind and equal. Reading the judgment bridge.

Function derivability vs. continuity:

Theorem: If the function f(x) is derivable at x0, it must be continuous at the point x0. The above Dingzi Meng's theory explains: the function can be derivative, and the function is continuous; Function continuity is not necessarily derivable; Discontinuous functions must not be derivative.

Derivable and differentiable equivalents in unary functions are independent of the integrable.

Multivariate functions can be differentiated, but not vice versa.

That is, in a univariate function, derivability is a sufficient and necessary condition for differentiability;

In multivariate functions, derivability is a necessary condition for differentiability, and differentiability is a sufficient condition for derivability.

In a univariate function, differentiability and derivability are equivalent.

In a multivariate function, the condition that a point is differentiable is that it is derivable in all directions.

Differentiability means that a curve can be divided into many infinitesimal fragments, and there are no breakpoints to derive, meaning that it is not only differentiable but also smooth.

Differentiation is not necessarily leadable, but can be adopted differently.