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Derivative of Inverse Function

Examples with detailed solutions on how to find the derivative (Differentiation) of an inverse function, in calculus, are presented.

Example 1: Find the derivative dy/dx of the inverse of function f defined by

f(x)= (1/2) x - 1


Solution to Example 1:

We present two methods to answer the above question

Method 1:

  • The first method consists in finding the inverse of f and differentiate it. To find the inverse of f we first write it as an equation

    y = (1/2) x - 1

  • Solve for x.

    x = 2y + 2.

  • Change y to x and x to y.

    y = 2x + 2.

  • The above gives the inverse function of f. Let us find the derivative

    dy / dx = 2


Method 2:

  • The second method starts with one of the most important properties of inverse functions.

    f(f -1(x)) = x

  • Let y = f -1(x) so that.

    f(y) = x.

  • Differentiate both sides using chain rule to the left side.

    (dy/dx)(df/dy) = 1.

  • Solve for dy/dx

    dy / dx = 1 / (df / dy)

  • f is defined by

    f(x)= (1/2) x - 1

  • so that df / dy = 1/2

  • Substitute df / dy by 1/2 in dy / dx = 1 / (df / dy) to obtain

    dy / dx = 1 / (1/2) = 2
  • The first method can be used only if we can find explicitly the inverse function.

    Example 2: Find the derivative dy / dx where y = arcsin x.

    Solution to Example 2:

    • arcsin x is the inverse function of sin x and

      sin(arcsin(x)) = x

    • y = arcsin x so that

      sin y = x

    • Differentiate both sides of the above equation, with respect to x, using the chain rule on the left side.

      dy/dx cos y = 1

    • Solve for dy/dx.

      dy/dx = 1 / cos y

      = 1 / cos ( arcsin x)

      = 1 / sqrt(1 - sin 2(arcsin x))

      = 1 / sqrt (1 - x 2)

    Exercises: Find the derivative of the inverse of each function.

    1 - f(x) = 3x - 4

    2 - g(x) = arccos x

    3 - h(x) = arctan x

    solutions to the above exercises

    1 - f -1 ' (x) = 1 / 3

    2 - g -1 ' (x) = -1 / sqrt(1 - x 2)

    3 - h -1 ' (x) = 1 / (1 + x 2)

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    Updated: 2 April 2013

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