Today we will consider computing the derivative of the inverse of a function. This is pretty standard and is in all the textbooks.
The usual suspects are the inverse trigonometric functions. So let’s start with and then rewrite this as . Differentiating this gives
Since we would like this in terms of x we can proceed two ways.
The denominator is the cosine of the number whose sine is x. So using the relationship
we find that
That tends to be confusing so another method is to draw a right triangle with an acute angle of y and arrange the side so that
A second example: Find the derivative of . The domain of this function is and the range is , the function is increasing on both parts of its domain; we will need to know this.
Proceeding as above we will find that
Drawing a triangle as above and arranging the side so that sec(y) = x:
But wait! It may be that x < 0, but is increasing and the derivative should always be positive. So this needs to be adjusted to
These can be a bit tricky.
Next: the fifth and last posting in this series will look at the graphical and numerical aspects of the derivatives of a function and its inverse.