derivative of inverse function

Theorem.  If the real function f has an inverse function f and the derivative of f at the point   x=f(y)  is distinct from zero, then f is also differentiableMathworldPlanetmathPlanetmath at the point y and

f(y)=1f(x). (1)

That is, the derivatives of a functionMathworldPlanetmath and its inverse function are inverse numbers of each other, provided that they have been taken at the points which correspond to each other.

{it Proof. Now we have


The derivatives of both sides must be equal:


Using the chain ruleMathworldPlanetmath we get




This is same as the asserted (1).

Examples.  For simplicity, we express here the functions by symbols y and the inverse functions by x.

  1. 1.

    y=tanx,  x=arctany;  dxdy=1dydx=11+tan2x=11+y2

  2. 2.

    y=sinx,  x=arcsiny;  dxdy=1dydx=1cosx=1+1-sin2x=+11-y2

  3. 3.

    y=x2,   x=±y;  dxdy=1dydx=12x=1±2y

If the variable symbol y in those results is changed to x, the results can be written

Title derivative of inverse function
Canonical name DerivativeOfInverseFunction
Date of creation 2015-02-21 16:02:46
Last modified on 2015-02-21 16:02:46
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Theorem
Classification msc 26A24
Related topic InverseFunctionTheorem
Related topic Derivative2
Related topic DerivativeOfTheNaturalLogarithmFunction
Related topic CyclometricFunctions
Related topic SquareRoot
Related topic LimitExamples
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