There are three formulas for the definition of derivative: the first formula f(x0)=limx→x0f(x)-f(x0)/(x-x0). The second formula f' (x0) = limh → 0f (x0+h)-f (x0)/H. The third formula f (x0) = lim δ x→ 0 δ y/δ x, and the relevant information is as follows:

1, derivative, also called derivative function, is one of the basic concepts in differential calculus. It reflects the rate of change of a function at a certain point, that is, the sensitivity of the function at that point.

2. The definition of derivative has several different forms, but the most basic one is the limit form. The first formula is the limit of the difference between functions f(x) and f(x0) and the ratio of x-x0 when the derivative is at point x0. When this limit exists, we say that the function f is differentiable at point x0.

3. It expresses the limit of the ratio of function f at point x0+h and the difference between x0 and h, when h approaches 0 from the right. If this limit exists, we say that the function f is differentiable at point x0.

4. The existence and continuity of derivatives are two important properties of functions. Whether the derivative exists depends on whether the slope of the function at each point is limited. If the slope of a function at a certain point is infinite, then the derivative at that point does not exist. The continuity of derivative means that the rate of change of the function at each point is continuous, and there is no jump or mutation.

Application of derivative

1, the maximum and extreme value of the function: the derivative can be used to find the maximum and extreme value of the function. By calculating the derivative of the function, we can find the point where the function grows fastest (maximum point) and the point where the function grows slowest (minimum point). In practical application, this kind of application is very common.

2. Tangents and normals of curves: Derivatives can be used to find the tangents and normals of curves. In two-dimensional graphics, the tangent of a curve is the slope of the curve at a certain point, and the normal is a straight line perpendicular to the tangent. In three-dimensional graphics, the normal of a surface is perpendicular to the surface. These concepts are widely used in the fields of geometry and graphic design.

3. Optimization problem: In many practical problems, we need to find the optimal solution that satisfies some constraints. For example, on issues such as road design, production planning and financial investment, we need to find the optimal decision to achieve maximum profit or minimum cost. Derivative can help us find the optimal solution, because it can reflect the monotonicity of the function and help us determine the location of the optimal solution.