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Derivatives of Inverse Functions

Our goal in this section is to come up with formulas for the derivatives of the inverse trigonometric functions. We could do this individually for each function but the process will be quicker if we can derive a general rule for finding the derivative of the inverse of a function in terms of the original functions inverse.

To do this suppose $g(x)$ and $f(x)$ are inverses of each other, i.e. $g(x) = f^{-1}(x)$. We're going to derive a formula for the derivative of the inverse in the upcoming lecture but the process is a relatively straightforward application of the Chain Rule so see if you can do this yourself first. Start with the fact that the composition of a function and its inverse is equal to $x$ then differentiate both sides of that equation using the Chain Rule. Remember that we're assuming you know the derivative of the original function, $f'(x)$ in this case, and that you're trying to find a formula for $g'(x)$ in terms of $f'(x)$.

Hint: Start with the equation $f(g(x)) = x$ and differentiate both sides using the Chain Rule.

Final Result: $g'(x) = \frac{d}{dy}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$

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