Derivation rule: given f (x) = (1 + x ^ 2) arctanx, find f '(0)

Derivation rule: given f (x) = (1 + x ^ 2) arctanx, find f '(0)

f'(x)=(1+x ²)'* arctanx+(1+x ²)* (arctanx)'
=2xarctanx+(1+x ²)* 1/(1+x ²)
=2xarctanx+1
So f '(0) = 0 + 1 = 1

(x ^ 2-3x + 1) ^ 3 how to take the third derivative,

y=(x^2-3x+1)^3
y'=3(x^2-3x+1)^2(x^2-3x+1)'
y'=3(x^2-3x+1)^2(2x-3)

What is the result of the derivative of X · x ^ e?

=（x）'X^e+x*(X^e)'
=X^e+xX^e
=(1+x)X^e

The derivative of e ^ x-e ^ (- x) is e ^ x + e ^ (- x), The derivative of e ^ x is followed by e ^ X. isn't it e ^ (- x) after the derivative of e ^ (- x)? How can it become - e ^ (- x)?

In order to obtain the derivative of composite function, the composite function must be decomposed into simple functions, and then the derivative and multiplication are obtained respectively
In your question, e ^ x is a simple function, but e ^ (- x) is not a simple function. It is composed of the function y = e ^ u and the function u = - x, so you can't directly use the formula you remember. The derivative of e ^ is e ^ x, and your derivative of e ^ (- x) becomes e ^ (- x), which is obviously wrong. The derivative of e ^ (- x) should be derived from e ^ u to get e ^ u, and then from - x to get - 1. They multiply to get - e ^ (- x)
That's your question. Have you just started learning this? For derivation, we must remember the basic form of the function. For example, the derivative of LN x is 1 / x, but ln x ^ 2 can not be simply regarded as 1 / x ^ 2, because it is a composite function composed of y = ln u and u = x ^ 2
So the answer is 1 / x ^ 2 times the derivative of x ^ 2 2x. The final result is 2 / X

Why is the original function of 1 / (COS squared x) TaNx? How can I calculate how negative TaNx is? Please give the process

(tanx)'
=(sinx/cosx)'
=(cos^2x+sin^2x)/cos^2x
=1/cos^2x

Known function f (x) = √ 3sinxcosx + cos square X-1 / 2, X ∈ R (1) Find the minimum positive period and monotone increasing interval of the function

f(x)=√3sinxcosx+cos ² X = (√ 3 / 2) sin2x + (1 + cos2x) / 2-1 / 2 = (√ 3 / 2) sin2x + (1 / 2) cos2x = sin2x * cos (π / 6) + cos2x * sin (π / 6) = sin (2x + π / 6) (1) t = 2 π / 2 = π increasing interval 2K π - π / 2 ≤ 2x + π / 6 ≤ 2K π + π / 22K π - 2 π / 3 ≤ 2x ≤ 2K π + π / 3