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Section2.12Inverse Trigonometric Functions

One very useful application of implicit differentiation is to find the derivatives of inverse functions. We have already used this approach to find the derivative of the inverse of the exponential function — the logarithm.

We are now going to consider the problem of finding the derivatives of the inverses of trigonometric functions. Now is a very good time to go back and reread Section 0.6 on inverse functions — especially Definition 0.6.4. Most importantly, given a function \(f(x)\text{,}\) its inverse function \(f^{-1}(x)\) only exists, with domain \(D\text{,}\) when \(f(x)\) passes the “horizontal line test”, which says that for each \(Y\) in \(D\) the horizontal line \(y=Y\) intersects the graph \(y=f(x)\) exactly once. (That is, \(f(x)\) is a one-to-one function.)

Let us start by playing with the sine function and determine how to restrict the domain of \(\sin x\) so that its inverse function exists.