Differential: dividing a function into infinitesimal parts. When the curve shrinks infinitely, it can be treated as a straight line, and the differential can be expressed as the product of derivative and dx. This is the direction that Leibniz put forward and studied.
In fact, there is no difference between derivative and differential in essence, just the difference in research direction.
Integral: definite integral is to find the area between curve and x axis; Indefinite integral is an equation satisfied by this area, so the latter is a means to find definite integral. Indefinite integral is essentially variable definite integral.
In other words:
Derivative y' is the rate of change of a function at a certain point, differential is the amount of change, and derivative is the quotient of function differential and independent variable differential, that is, y'=dy/dx, so the theories and methods of derivative and differential are collectively called differential calculus (knowing the function, finding derivative or differential). Integral is the inverse problem of differential calculus.
Limit is the basis of differential, derivative, indefinite integral and definite integral. When Newton and Leibniz first discovered calculus, there was no strict definition. Later, French mathematician Cauchy used limit to make calculus have a strict mathematical foundation. Limit is the basis of derivative, and derivative is the simplification of limit. Differential is the deformation of derivative.
Differential: the increment of infinite blocks can be regarded as the rate of change, that is, the derivative. Integral: The sum of the areas of infinite blocks can be regarded as the whole area.
Derivative must be continuous, continuity on closed interval must be integrable, and integrable must be bounded.
derivant
Derivative is an important basic concept in calculus. When the independent variable x of the function y=f(x) generates an increment δ x at the point x0, if there is a limit a in the ratio of the increment δ y of the function output value to the increment δ x of the independent variable when δ x tends to 0, then A is the derivative at x0, which is denoted as f'(x0) or df(x0)/dx. 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. Not all functions have derivatives, and a function does not necessarily have derivatives at all points. If the derivative of a function exists at a certain point, it is said to be derivative at this point, otherwise it is called non-derivative. However, the differentiable function must be continuous; Discontinuous functions must be non-differentiable.
For differentiable function f(x), x? F'(x) is also a function called the derivative function of f(x). The process of finding the derivative of a known function at a certain point or its derivative function is called derivative. Derivative is essentially a process of finding the limit, and the four algorithms of derivative also come from the four algorithms of limit. Conversely, the known derivative function can also reverse the original function, that is, indefinite integral. The basic theorem of calculus shows that finding the original function is equivalent to integral. Derivation and integration are a pair of reciprocal operations, both of which are the most basic concepts in calculus.