Inverse functions and differentiation
The inverse of a function  is a function that, in some fashion, "undoes" the effect of  (see inverse function for a formal and detailed definition). The inverse of is denoted  . The statements y=f(x) and x=f-1(y) are equivalent.
Differentiation in calculus is the process of obtaining a derivative. The derivative of a function gives the slope at any point.
 denotes the derivative of the function  with respect to  .
 denotes the derivative of the function  with respect to  .
The two derivatives are, as the Leibniz notation suggests, reciprocal, that is
This is a direct consequence of the chain rule, since
and the derivative of with respect to is 1.
Examples
-
 (for positive ) has inverse  .
-
 has inverse  (for positive ).
Additional properties
- Integrating this relationship gives
- This is only useful if the integral exists. In particular we need
 to be non-zero across the range of integration.
- It follows that functions with continuous derivative have inverses in a neighbourhood of every point where the derivative is non-zero. This need not be true if the derivative is not continuous.
Related Topics
calculus, inverse functions, chain rule
Referenced By
Derivative | Derivative (calculus) | Differentation | Differentiable function | Inverse | List of calculus topics | List of mathematical topics (G-I) | List of mathematical topics (G-Z)
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