( 1-y)/( 1+y)

( 1+x)y= 1-x，xy+x= 1-y，x( 1+y)= 1-y，x=( 1-y)/( 1+y)

x/4

y=4x→x=y/4→y^(- 1)=x/4

e^(sin？ x)

y=e^(sin？ t)=e^(sin？ x)

y=2^x/( 1+2^x)

2^y( 1-x)=x,2^y-2^y*x=x,( 1+2^y)*x=2^y,x=2^y/( 1+2^y)

When writing the inverse function here, X and Y cannot be interchanged, that is, the inverse function of y=arctanx is x=tany.

So x=tany, dx/dy=(tany)'=sec2y?

dy/dx = 1/(dx/dy)= 1/sec2y = cos2y/(cos2y+sin2y)= 1/( 1+sin2y/cos2y)= 65438+。

Related definitions:

Generally speaking, let the range of function y=f(x)(x∈A) be C. If there is function g(y) everywhere, such function x = g(y) (y ∈ c) is called function y = f (x) (x ∈ inverse function x = f-/kl. The most representative inverse functions are logarithmic function and exponential function.

Generally speaking, if X and Y correspond to a corresponding relation f(x) and y=f(x), the inverse function of y=f(x) is x=f- 1(y). The condition for the existence of inverse function (single-valued function by default) is that the original function must be in one-to-one correspondence (not necessarily in the whole number domain). Note: superscript "? 1 "refers to the function power, not the exponential power.