Derivatives of Inverses

Properties of Inverses:

1. f(f ^-1(x)) = f ^-1(f(x)) = x
2. A function and its inverse are symmetric to each other over the line y = x
3. If (a, b) is on f(x), then (b, a) is on f ^-1(x)
4. If f(x) passes the horizontal line test, then its inverse is also a function
5. A function has an inverse if it is 1-1
6. If a function is strictly monotonic, then it is 1-1 and has an inverse function
1. To see if a function is strictly monotonic, check its derivative
7. If f(x) is continuous, then f ^-1(x) is continuous
8. If f(x) is increasing, then f ^-1(x) is increasing; also, if f(x) is decreasing, then f ^-1(x) is decreasing
9. If f(x) is differentiable at x = c and f(c) is not equal to 0, then f ^-1(x) is differentiable at
   x = f(c)
   

Derivative of an Inverse Function:

Let f ^-1(x) = g(x)           Then f(g(x)) = x
                                 f '(g(x))(g'(x)) = 1   (Take the derivative of both sides)
                                                g'(x) =      1      
                                                            f'(g(x))

      Example:   

1. Find g'(x) if f(x) = 2x² - 1 and g(x) = f ^-1(x)
1. g(x) = \/((x + 1)/2)  (Interchange x and y in y = 2x² - 1 and solve for y)
2. f'(x) = 4x
3. g'(x) =      1        =            1                  =              1            =         \/2       
                f'(g(x))        f'(\/((x + 1)/2))             4(\/((x + 1)/2)        4\/(x + 1)
   
2. Find g'(7)
1.      \/2          =       \/2      =     1    =    1
   4\/(7 + 1)            4\/8          4\/4        8
   
3. Find g'(7) using only the points and the derivative formula
1. g'(7) =      1     
               f'(g(7))
2. f(x) = 2x² - 1
   If 7 is an x value for g(x), then 7 is a y value for f(x)
3.      7 = 2x² - 1
        8 = 2x²
        4 = x²
    2,-2 = x
4. g'(7) =    1     =      1      
               f'(2)           8

Problems