Differentiable at x, both left and right derivatives exist and are equal

slope

Velocity is the derivative of displacement with respect to time, the derivative of electric quantity with respect to time is current intensity, and the derivative of mass with respect to lines and planes

Compound derivation

Derivation of functions determined by parametric equations

higher order derivatives

Differentiable must be continuous, continuous may not be differentiable (corner points are not differentiable)

Necessary and sufficient conditions, derivable at x

Partial derivative

Total differential

Composite multivariate function

Implicit function derivation rule

The derivative is greater than 0 and increases monotonically.

Rolle's theorem

Lagrange's mean value theorem

Judgment of extreme value

best value

Extreme value of multivariate small function

Concave and convex

inflection point

There are both left and right limits

At least one of the left and right limits does not exist

Elementary functions are continuous within a defined interval

There must be a maximum and minimum value

There must be a boundary

zero point theorem

Intermediate value theorem

The sum of a finite number of infinitesimal algebras is infinitesimal, but an infinite number may not

The product of bounded functions (limited variables, constants, finite infinitesimals) and infinitesimals is infinitesimal,

infinitesimal comparison