Inverse function rule

In <a href="/facts/Calculus/MEmUTYoz">calculus</a>, the inverse function rule is a <a href="/facts/Formula/BosTcKnv">formula</a> that expresses the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of the <a href="/facts/Inverse_function/Tg5nnXlu">inverse</a> of a <a href="/facts/Bijective/j4gTuTmW">bijective</a> and <a href="/facts/Differentiable_function/jQqTgLk1">differentiable function</a> f in terms of the derivative of f. More precisely, if the inverse of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is denoted as 
 
 
 
 
 f
 
 −
 1
 
 
 
 
 {\displaystyle f^{-1}}
 
, where 
 
 
 
 
 f
 
 −
 1
 
 
 (
 y
 )
 =
 x
 
 
 {\displaystyle f^{-1}(y)=x}
 
 if and only if 
 
 
 
 f
 (
 x
 )
 =
 y
 
 
 {\displaystyle f(x)=y}
 
, then the inverse function rule is, in <a href="/facts/Lagrange%2527s_notation/PTCbwS0Y">Lagrange's notation</a>,

[
 
 f
 
 −
 1
 
 
 ]
 
 ′
 
 (
 y
 )
 =
 
 
 1
 
 
 f
 ′
 
 
 (
 
 
 f
 
 −
 1
 
 
 (
 y
 )
 
 )
 
 
 
 
 
 
 {\displaystyle \left[f^{-1}\right]'(y)={\frac {1}{f'\left(f^{-1}(y)\right)}}}
 
.
This formula holds in general whenever 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> and <a href="/facts/Injective_function/5ES6tqf5">injective</a> on an interval I, with 
 
 
 
 f
 
 
 {\displaystyle f}
 
 being differentiable at 
 
 
 
 
 f
 
 −
 1
 
 
 (
 y
 )
 
 
 {\displaystyle f^{-1}(y)}
 
(
 
 
 
 ∈
 I
 
 
 {\displaystyle \in I}
 
) and where
 
 
 
 
 f
 ′
 
 (
 
 f
 
 −
 1
 
 
 (
 y
 )
 )
 ≠
 0
 
 
 {\displaystyle f'(f^{-1}(y))\neq 0}
 
. The same formula is also equivalent to the expression

D
          
        
        
          [
          
            f
            
              −
              1
            
          
          ]
        
        =
        
          
            1
            
              (
              
                
                  D
                
              
              f
              )
              ∘
              
                (
                
                  f
                  
                    −
                    1
                  
                
                )
              
            
          
        
        ,
      
    
    {\displaystyle {\mathcal {D}}\left[f^{-1}\right]={\frac {1}{({\mathcal {D}}f)\circ \left(f^{-1}\right)}},}

where 
 
 
 
 
 
 D
 
 
 
 
 {\displaystyle {\mathcal {D}}}
 
 denotes the unary derivative operator (on the space of functions) and 
 
 
 
 ∘
 
 
 {\displaystyle \circ }
 
 denotes <a href="/facts/Function_composition/r5Pvnmth">function composition</a>.
Geometrically, a function and inverse function have <a href="/facts/Graph_of_a_function/htYX0BGX">graphs</a> that are <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflections</a>, in the line 
 
 
 
 y
 =
 x
 
 
 {\displaystyle y=x}
 
. This reflection operation turns the <a href="/facts/Slope/a2L3k0j5">gradient</a> of any line into its <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocal</a>.
Assuming that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has an inverse in a <a href="/facts/Neighborhood_(mathematics)/XHKgrpkp">neighbourhood</a> of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and have a derivative given by the above formula.
The inverse function rule may also be expressed in <a href="/facts/Leibniz%2527s_notation/FLytWNf9">Leibniz's notation</a>. As that notation suggests,

d
              x
            
            
              d
              y
            
          
        
        
        ⋅
        
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        =
        1.
      
    
    {\displaystyle {\frac {dx}{dy}}\,\cdot \,{\frac {dy}{dx}}=1.}

This relation is obtained by differentiating the equation 
 
 
 
 
 f
 
 −
 1
 
 
 (
 y
 )
 =
 x
 
 
 {\displaystyle f^{-1}(y)=x}
 
 in terms of x and applying the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a>, yielding that:

d
              x
            
            
              d
              y
            
          
        
        
        ⋅
        
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        =
        
          
            
              d
              x
            
            
              d
              x
            
          
        
      
    
    {\displaystyle {\frac {dx}{dy}}\,\cdot \,{\frac {dy}{dx}}={\frac {dx}{dx}}}

considering that the derivative of x with respect to x is 1.

Inverse function rule open-in-new

Inverse function rule