Derivative of Inverse functions

We know the formula for finding the derivative
dy/dx = lim [f(x + δx) - f(x)]/δx
       δx→0

f(x) and f-1(y) are inverse functions.
y = f(x)
x = f-1(y)
(y + δy) = f(x + δx)
x + δx = f-1(y + δy)


dx/dy = lim [f-1(y + δy) - f-1(y)]/δy
       δy→0
 
(dy/dx)(dx/dy) = {lim [f(x + δx) - f(x)]/δx}{lim [f-1(y + δy) - f-1(y)]/δy}
                  δx→0                       δy→0
(dy/dx)(dx/dy) = {lim [y + δy - y]/δx}{lim [x + δx - x]/δy}
                  δx→0                       δy→0 
(dy/dx)(dx/dy) = {lim δy/δx}{lim δx/δy}
                  δx→0       δy→0  
(dy/dx)(dx/dy) = lim  lim {(δy/δx)(δx/δy)}
                 δx→0 δy→0  
               = lim  lim 1 = 1
                 δx→0 δy→0  
To find the inverse functions,
if y = f(x) then x = f-1(y) or
if x = g(y) then y = g-1(x) or

use dy/dx = 1/(dx/dy)



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