What is the derivative of the tangent function, and what is the derivative formula of the arctangent

The derivative of the tangent function is (secx) 2;

Derivatives are good local properties of functions. The derivative of a function at a point describes the rate of change of the function around that point. If both the independent variables and the values of the function are real, the derivative of the function at a point is the tangent slope of the curve represented by the function at that point.

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

The details are as follows: tan x).'=sin x /cos x)'

sin x)'cos x-sin x(cos x)']cosx*cos x

cos x*cos x-(-sin x*sin x)]/cos x*cos x

1/cos x*cos x

sec x*sec x

Significance of Derivatives:

If the function y=f(x) is in the open interval.

Each point in is derivative, and the function y=f(x) corresponds to a definite derivative for each definite x value in the interval, and the derivative is calculus.

An important pillar of the early clan of Lu Yusan.

The function y=f(x) is the derivative f at the point x0'The geometric meaning of (x0) represents the tangent of the curve of the function at the point p0(x0,f(x0)).

(the geometric meaning of the derivative stop is the tangent slope of the curve at this point).

arctanx)'The inverse function of the radical brigade = 1 (1 + x 2) function y = tanx, (x is not equal to k + 2, k z).

Written as x=arctany, it is called the arctangent function.

Its value range is (Bright Lead Stool - 2, 2). The arctangent function is an inverse trigonometric function.

one.

sec(arctanx)=√x^2+1)。

The specific calculation process is shown in the following figure:

The first derivative of the positive ant-sensitive cut-in function tanx is sec 2(x). In order to find its nth derivative, we can use the method of repeated derivatives, i.e., take the derivative of the (n-1) derivative of tanx.

Assuming that the n-1 derivative of tanx is f(n-1)(x), then its nth derivative is:

f(n)(x) =d/dx[f(n-1)(x)] d/dx[sec^2(x)f(n-2)(x)] sec^2(x)f(n-1)(x) +2sec^2(x) f(n-2)(x)

where f(0)(x) = tanx and f(1)(x) = sec 2(x) are the first and second derivatives of the tangent function. Therefore, we can use the above formula to solve repeatedly until we get the required n derivative.

For example, when n=2, we have:

f(2)(x) =sec^2(x)f(1)(x) +2sec^2(x)f(0)(x) =sec^4(x) +2tanx sec^2(x)

Therefore, the second derivative of tanx is sec 4(x) +2tanx sec 2(x).

Similarly, we can use the above formula to find the third derivative, fourth derivative, and so on of tanx, and suffocate the stool branch until the required coarse nth derivative.

arctanx)'The inverse function of the radical brigade = 1 (1 + x 2) function y = tanx, (x is not equal to k + 2, k z).

Written as x=arctany, it is called the arctangent function.

Its value range is (Bright Lead Stool - 2, 2). The arctangent function is an inverse trigonometric function.

one.

tan x )'sin x /cos x)'

sin x)'cos x-sin x(cos x)']cosx*cos x

cos x*cos x-(-sin x*sin x)]/cos x*cos x

1/cos x*cos x

sec x*sec x does not have derivatives for all functions, and a function does not necessarily have derivatives at all points. If a function exists at a certain point in derivative, it is said to be derivable at that point, otherwise it is called underivable. However, leadable'The function must be continuous; Discontinuous functions must not be derivative.

For derivative functions f(x), xf'(x) is also a function called a derivative of f(x). The process of finding the derivative of a known function at a point or its derivative is called derivative. In essence, derivation is a process of finding the limit, and the four rules of operation of derivatives are also the same as the four rules of operation of the limit.

Conversely, a known derivative can also be reversed to find the original function, i.e., an indefinite integral. The fundamental theorem of calculus states that the original function is equivalent to the integral. Derivative and integration are a pair of reciprocal operations, and they are both the most fundamental concepts in calculus.