Derivative is the local property of function. The derivative of a function at a certain point describes the rate of change of the function near that point. If the independent variables and values of the function are real numbers, then the derivative of the function at a certain point is the tangent slope of the curve represented by the function at that point.

The essence of derivative is the local linear approximation of function through the concept of limit. For example, in kinematics, the derivative of the displacement of an object with respect to time is the instantaneous velocity of the object.

2. Geometric significance:

The geometric meaning of the derivative f'(x0) of the function y=f(x) at x0: it represents the tangent slope of the function curve at P0(x0, f(x0)) (the geometric meaning of the derivative is the tangent slope of the function curve at this point).

3. Function:

Derivative is closely related to physics, geometry and algebra: tangent can be found in geometry; Instantaneous rate of change can be found in algebra; Speed and acceleration can be found in physics.

Derivative, also known as epoch number and WeChat quotient (the concept in differentiation), is a mathematical concept abstracted from the problems of speed change and tangent of curve (the direction of vector speed), also known as rate of change.

Extended data:

I. Calculation of Derivative

The derivative function of a known function can be calculated by using the limit of change rate according to the definition of derivative. In practical calculation, most common analytic functions can be regarded as the result of sum, difference, product, quotient or mutual compound of some simple functions. As long as the derivative functions of these simple functions are known, the derivative functions of more complex functions can be calculated according to the derivative law.

Second, the nature and function of derivatives

1, monotonicity

(1) If the derivative is greater than zero, it will increase monotonically; If the derivative is less than zero, it decreases monotonically; The derivative equal to zero is the stagnation point of the function, not necessarily the extreme point. To judge monotonicity, the derivatives of the left and right values of the entry point are required.

(2) If the known function is increasing function, the derivative is greater than or equal to zero; If the known function is a subtraction function, the derivative is less than or equal to zero.

2. Bump

The concavity and convexity of differentiable function is related to the monotonicity of its derivative. If the derivative function of a function increases monotonically in a certain interval, then the function in this interval is concave downward, otherwise it is convex upward.

If the second derivative function exists, it can also be judged by its positive and negative. If it is always greater than zero in a certain interval, the function is concave downward in this interval and convex upward in this interval. The concave-convex boundary point of a curve is called the inflection point of the curve.

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