what is the derivative of ?
Not necessarily. With the inverse trig functions, a very common procedure to find the derivative of the inverse trig functions is to use implicit differentiation. This is how, for example,
is done.
You have implying that
Implicitly differentiating yields
Now, if you draw the correct triangle, you will see that
(Or you can just use the Pythagorean theorem.)
Does that help?
Also, just in case a picture helps...
... where (key in spoiler) ...
Spoiler:
Here is how you could also allow for an inner function like 1/y.
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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