Given the function f (x) = (1 + x) ^ 2-ln (1 + x) ^ 2 (1) find the monotone interval of function f (x) (2) if function f (x) and function g (x) = x ^ 2 + X + a are in the interval [0,2] Sorry, I forgot to read the word limit when typing the second question, (2) If function f (x) and function g (x) = x ^ 2 + X + a have exactly two different intersections on interval [0,2], find the value range of real number a

Given the function f (x) = (1 + x) ^ 2-ln (1 + x) ^ 2 (1) find the monotone interval of function f (x) (2) if function f (x) and function g (x) = x ^ 2 + X + a are in the interval [0,2] Sorry, I forgot to read the word limit when typing the second question, (2) If function f (x) and function g (x) = x ^ 2 + X + a have exactly two different intersections on interval [0,2], find the value range of real number a

Here's your analysis: 1) the definition field of F (x) is (- OO, - 1) U (- 1, + OO), find the derivative f '(x) = 2 (x + 1) - 2 / (x + 1) = 2x (x + 2) / (x + 1), and get - 2 from F' (x) > 0

Given the function f (x) = ln (1 + x) - X / (1 + x), find the minimum of F (x)

f(x)=ln(1+x)-x/(1+x)
f'(x)=x/(x+1)^2
Let f '(x) = 0 get x = 0
Therefore, when x = 0, there is a minimum value, which is brought in to find f (x) = 0

lna*lnb*lnc=?

lna*lnb*lnc=ln((a^b)^c)
It's the b-th power of a, and then use this result as the base to the c-th power, and then find the natural logarithm

Derivation to the LN (TaNx) power of 2

Firstly, 2 ^ ln (TaNx) is an exponential composite function, the exponent ln (TaNx) itself is a logarithmic function, and ln (TaNx) contains tangent trigonometric function TaNx. Therefore, the derivation of it should first use the derivation formula of composite function: let the composite function y = f (g (x)), then its derivative number is y '= f' (g (x)) * g '(x) and refers to

The first partial derivative of Z = ln (TaNx / y) Find the total differential of Z = arctanx + Y / X-Y

(1) z=ln(tanx/y)
dz/dx=1/(tanx/y)*(sec ² x/y)=sec ² x/tanx
dz/dy=1/(tanx/y)*(-tanx/y ²)=- 1/y
(2) z=arctanx+y/x-y
dz=(dz/dx)*dx+(dz/dy)*dy
=[1/(x ²+ 1)-y/x ²] dx+(1/x-1)dy

Y = ln (TaNx / 2) derivation?

y=ln(tanx/2)
y'=1/tan(x/2)*sec^2(x/2)*(1/2)
=1/sinx