Concept: dy=Ax Δx(Δx→0)

Geometric meaning: Let Δx be the increment of point M on the curve y = f(x) on the abscissa, Δy be the increment of the curve on the ordinate corresponding to Δx at point M, and dy be the tangent line of the curve at point M. Corresponds to the increment of Δx on the ordinate.

differential formula

Definition: Suppose the function y=f(x) is defined in a certain neighborhood of point x0. When the independent variable x has an increment Δx at x0 and (x0 Δx) is also in this neighborhood, the corresponding function obtains an increment. Δy=f(x0 Δx)-f(x0); if the ratio of Δy to Δx, the limit exists when Δx→0, then the function y=f(x) is said to be differentiable at point x0. Not all functions have derivatives, and a function does not necessarily have derivatives at all points.

Geometric meaning: the slope of the tangent line passing through the tangent point of the curve.

Operation rules: including four arithmetic operations, derivation of composite functions and derivation of implicit functions

Derivative formula

higher order derivatives

Derivatives of functions determined by implicit functions and parametric equations

Derivative applications

Continuous may not be differentiable

Derivative must be continuous

Differentiable must be conductive

The linear principal part of a solvable function

Special points about solvable functions

Characteristics of graphs that can be roughly judged about functions

Ability to create function graphs more accurately

Can solve speed and acceleration in physics, etc.

Able to prove some theorems, rules, etc.

Can solve some optimization problems in life, find the optimal value, etc.

Utilize the relationship between differentiability, derivability, and continuity

Proficient in derivation operations of higher-order derivatives

Use the intermediate value theorem, the maximum value theorem, etc. to perform relevant proofs

Utilizing the characteristics of function monotonicity and concavity