Derivative of Inverse Function

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

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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 function 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 / √(1 - sin 2(arcsin x))
    = 1 / √ (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 / √(1 - x 2)
3) h -1 ' (x) = 1 / (1 + x 2)

More References and links

Inverse Function
differentiation and derivatives