Definition: make, and then
note:
1. The derivative is the limit of the ratio of function increment to independent variable increment. This incremental ratio is called the average rate of change (difference quotient) of the function relative to the independent variable, and the derivative is the rate of change of f relative to X.
2. If the limit of the increment ratio does not exist, it is said that f is not derivable at this point.
Let f(x) be differentiable, then it is an infinitesimal quantity at that time, so it is called the finite increment formula of f(x) at this point.
Note: The formula pair is still valid.
Theorem: If the function F is differentiable at one point, then F is continuous at one point.
Note: the derivative must be continuous, and continuous is not necessarily derivative.
Example: Prove that the function is only differentiable in point, here is Dirichlet function.
Certificate:
Definition: Let a function be defined in the right neighborhood of a point. If there is a right limit, the limit value is called the right derivative of f at this point, and it is recorded as
Similar definition of left derivative
The right derivative and the left derivative are collectively called unilateral derivatives.
Theorem: If a function is defined in the neighborhood of a point, then both exist and
Definition: If every point of a function on the interval I is differentiable (only the corresponding one-sided derivative is considered at the end of the interval), then F is called a differentiable function on I, and at this time, for each one, there is a derivative (or one-sided derivative) of F corresponding to it, thus defining a function on I, which is called the derivative function of F on I for short.
that is
note:
1. The derivative y' in physics is also often marked by Newton.
2. Sometimes writing or
Example: proof (sinx)'=cosx
Certificate:
Example: prove, especially
Certificate:
Tangent equation of curve at point
The derivative of function f at point is the tangent slope of the curve at point.
Indicates the angle between the tangent and the positive direction of the X axis, and then
Example: Find the tangent equation and normal equation of the curve at this point.
Solution:
Note: For curves, the tangent slope of this point can be rewritten as follows:
Therefore, in order to make a tangent to point P, we can divide the line segment from the origin O to the point on the X axis in half, and take the near bisector Q, then the straight line PQ is the tangent.
Definition: If there is a function f, it is said that the function f obtains a maximum (minimum) value at a point, which is called the maximum (minimum) value point.
The maximum and minimum values are collectively referred to as extreme values, and the maximum and minimum values are collectively referred to as extreme points.
Example: Prove: If, then, yes.
Certificate:
Note: If it exists and is not zero, it is not the extreme point of f(x).
Theorem: Let the function f be defined at one point and be derivable at one point. If the point is the extreme point of f, then
Geometric meaning: If the function is differentiable at the extreme point, then the tangent of this point is parallel to the X axis.
The point that satisfies the equation is called the stable point.
For example, for a function, point x=0 is a stable point, but not an extreme point.