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How to Differentiate Inverse Functions

There’s a difficult-looking formula involving the derivatives of inverse functions, but before you get to that, look at the following figure, which nicely sums up the whole idea.

image0.jpg

This figure shows a pair of inverse functions, f and g. Inverse functions are symmetrical with respect to the line, y = x. As with any pair of inverse functions, if the point (10, 4) is on one function, (4, 10) is on its inverse. And, because of the symmetry of the graphs, you can see that the slopes at those points are reciprocals:

image1.png

That’s how the idea works graphically.

The algebraic explanation is a bit trickier, however:

  1. Write the point (10, 4) on f as (10, f (10)).

  2. Express the slope at this point (which is the derivative) as

    image2.png
  3. Write the point (4, 10) on g as (4, g (4)).

  4. Because f (10) = 4, replace the 4s in (4, g (4)) with f (10)s.

    This gives you (f (10), g (f (10))).

  5. Express the slope (the derivative) at this point, as

    image3.png

    This difficult-looking equation expresses nothing more and nothing less than the two triangles on the two functions in the preceding figure.

    image4.png

In words, this formula says that the derivative of a function, f, with respect to x, is the reciprocal of the derivative of its inverse with respect to f.

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