Then:

y=∫xdx

=( 1/2)x^2+c.

Therefore, the derivatives of (1/2) x 2+c are all x.

derivative

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.

definition

Let the function y=f(x) be defined in the neighborhood of point x0. When the independent variable x has an increment Δ x at x0 and (x0+Δ x) is also in the neighborhood, the corresponding function obtains the increment Δ y = f (x0+Δ x)-f (x0); If the ratio of Δ y to Δ x has a limit when Δ x→ 0, then the function y=f(x) is derivable at point x0, and this limit is called the derivative of the function y=f(x) at point x0, and is recorded as

derivative

If the function y=f(x) is differentiable at every point in the open interval, it is said that the function f(x) is differentiable in the interval. At this time, the function y=f(x) corresponds to a certain derivative value for every certain value of x in the interval, and forms a new function, which is called the derivative function of the original function y=f(x), and is abbreviated as y', f'(x), dy/dx or df(x)/dx.

Derivative is an important pillar of calculus. Newton and Leibniz contributed to this.

Geometric meaning

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).

formula

Simple function

Here we will enumerate the derivatives of the basic elementary function of 14.

function of a complex variable

Four operations of 1 and its derivatives;

2. The relationship between the original function and the derivative of the inverse function (the inverse trigonometric function is derived from the derivative of trigonometric function):

The inverse function of y=f(x) is x=g(y), so there is y'= 1/x'.

3, the derivative of the composite function:

The derivative of compound function to independent variable is equal to the derivative of known function to intermediate variable, multiplied by the derivative of intermediate variable to independent variable (called chain rule).

4, the derivative rule of variable limit integral:

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.

Derivation rule of derivative

The derivative function of a function composed of the sum, difference, product, quotient or mutual combination of basic functions can be derived from the derivative rule of the function. The basic deduction rules are as follows:

1, Linearity of Derivation: Finding the linear combination of derivative function is equivalent to finding the derivative of each part first, and then finding the linear combination (i.e. Formula ①).

2. Derivative function of the product of two functions: one derivative times two+one derivative times two (i.e. Formula ②).

3. The derivative function of the quotient of two functions is also a fraction: (derivative times mother-derivative times mother) divided by mother square (i.e. Formula ③).

4. If there is a compound function, use the chain rule to deduce it.

higher derivative

Solution of higher derivative

1. Direct method: Find the higher derivative step by step from its definition.

Generally used to find solutions to problems.

2. The algorithm of higher derivative:

3. Indirect method: using the known higher-order derivative formula, through four operations, variable substitution and other methods.

Note: the function after substitution should be easy to find, and try to get the order derivative as close as possible to the known formula.

Concise memory formula

In order to facilitate memory, someone has compiled the following formula:

Usually zero, power off

Reciprocal (direct reciprocal when E is the bottom, and multiply by 1/lna when A is the bottom)

The exponent is unchanged (especially the exponential function of natural logarithm is completely unchanged, and the general exponential function must be multiplied by lna).

Turn positive into profit, and profit is positive.

Tangent square (tangent function is the square of the corresponding tangent function (reciprocal of tangent function))

Cutting, Multiplication and Cutting, Inverse Fraction