Geometric meaning of derivative: For differentiable functions, tangent is infinitely approximated by secant, and the polar line of secant slope is the slope of tangent. The formula is: the derivative f'(x0) of the function y=f(x) at x=x0, which represents the slope k of the tangent of the curve y=f(x) at point P(x0, f(x0)). Derivative is an important basic concept in calculus.

If the function y = f(x) is differentiable at every point in the open interval I, it is said that the function f(x) is differentiable in the interval I. At this time, the function y = f(x) corresponds to a certain derivative of each certain value of x in the interval I, and constitutes a new function. The derivative function of the original function y = f(x) is called y'; F'(x), dy/dx, df(x)/dx, derivative function is called derivative for short.

Derivative definition

The first definition of derivative

Let the function y=f(x) be defined in the neighborhood of point x0. When the independent variable x has an increment △x at x0 (x0+△ x is also in the neighborhood), the corresponding function gets an increment △y=f(x0+△x)-f(x0). If the ratio of △y to △x exists when △ x→0, the function y = is called.

The second definition of derivative

Let the function y=f(x) be defined in the neighborhood of point x0. When the independent variable x changes △x at x0 (X-x0 is also in the neighborhood), the function changes △y=f(x)-f(x0) accordingly. When △x→0, if there is a limit in the ratio of △y to △ x, the function y=f(x) exists.