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We'll approach this topic via four examples, making good use of implicit differentiation:
And that's pretty much the graphical version of the Preview!:)
(WARNING: don't be fooled by the notation: $f^{-1}(x)$ is the inverse function; $f(x)^{-1}$is the multiplicative inverse, $\frac{1}{f(x)}$. They are completely different animals!)
\[ f(x)=x^2 \] and \[ f^{-1}(x)=\sqrt{x} = x^{\frac{1}{2}} \]
This is an interesting place to start, because the square function isn't actually invertible!
By convention, we restrict the domain, and choose to consider the square function on only half of its domain (positive values of $x$).
And you can see what goes wrong if we try to reflect the part to the left of the $y$-axis: the inverse "function" would fail the vertical line test!
We simply use implicit differentiation on those relationships above (in particular the first):
\[ \frac{d}{dx}\left(f(f^{-1}(x))\right) = \frac{d}{dx}(x) \] or (using the chain rule on the left) \[ f'(f^{-1}(x)) \frac{d}{dx}\left(f^{-1}(x)\right) = 1 \] So \[ \frac{d}{dx}\left(f^{-1}(x)\right) = \frac{1}{f'(f^{-1}(x))} \] Let's see what that produces in this case: for $f(x)=x^2$, $f'(x)=2x$, so \[ \frac{d}{dx}\left(f^{-1}(x)\right) = \frac{1}{2f^{-1}(x)} = \frac{1}{2\sqrt{x}} = \frac{1}{2}x^{-\frac{1}{2}} \]
So \[ \frac{d}{dx}\left(f^{-1}(x)\right) = \frac{1}{f'(f^{-1}(x))} \] becomes \[ \frac{d}{dx}\left(\ln(x)\right) = \frac{1}{e^{\ln(x)}} = \frac{1}{x} = x^{-1} \] Amazing! This is the "missing power" -- we didn't know how to get this particular derivative using the power rule! (It would have come from the power $x^0$ -- but that's a constant!)
Notice that it's our friend the hyperbola from last time. Notice also that this function is odd -- it should be the derivative of an even function. We can extend the $\ln(x)$ function to the left, by considering $\ln(|x|)$:
Look at sine on this interval. If it is increasing everywhere, its inverse must be increasing everywhere: and what does that say about its derivative?
There are two other functions which deserve our attention. We call their inverses "arctan" and "arccos".
Do exactly the same thing that we've just done in these three examples, but with two other important functions: