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# Derivative of inverse cosine

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.E (LO)
,
FUN‑3.E.2 (EK)

## Video transcript

in the last video we showed or we proved to ourselves that the derivative of the inverse sine of X is equal to 1 over the square root of 1 minus x squared what I encourage you to in this video is deposit and try to do the same type of proof for the derivative of the inverse cosine of X so what is so our goal here is to figure out I want the derivative with respect to X of the inverse cosine inverse cosine of X what is this going to be equal to so assuming you've had a go at it let's work through it so just like last time we could write let's just just set Y being equal to this Y is equal D inverse cosine of X which means the same thing is saying that X is equal to the cosine cosine of Y now let's take the derivative of both sides with respect to X on the left hand side you're just going to have a 1 we're just going to have a 1 and on the right hand side you're going to have the derivative of cosine Y with respect to Y which is negative sine of Y times the derivative of Y with respect to X which is DS dy/dx and so we get if we let's see if we divide both sides by negative sine of Y we get dy/dx is equal to negative 1 over sine of Y now like we've seen before this is kind of satisfying but we have our derivative in terms of Y we want it in terms of X and we know that X is cosine of Y so let's see if we can rewrite this bottom expression in terms of cosine of Y said of sine of Y well we know when we saw in the last video that from the Pythagorean identity that that cosine squared of Y plus sine squared of Y is equal to 1 we know that sine of Y is equal to the square root of 1 minus cosine squared of Y so this is equal to negative 1 this is just a manipulation of the Pythagorean trig identity this is equal to 1 minus cosine and I could write like this cosine squared of Y but I'll write it like this because it'll make it a little bit clearer and what is cosine of Y well of course that is X so this is equal to negative 1 over the square root of 1 minus instead of writing cosine y instead of writing cosine Y you're trying to switch colors instead of writing cosine Y we could write 1 minus X 1 minus x squared so there you have it the derivative with respect to X of the inverse cosine of X is is well I think I lost that color I'll do it in magenta is equal to 1 is equal to negative 1 over the square root of 1 minus x squared so this is a neat thing this right over here is a neat thing to know and of course we should compare it to the inverse the in the derivative of the inverse sine actually let me put them side by side and we see that the only difference here is is the sine so let me copy and paste that so copy and paste it I'm going to paste it down here and now let's look at them side by side so we see for taking the derivative with respect to X of the inverse cosine function we have a negative of a negative 1 over the square root of 1 minus x squared if you're looking at the derivative with respect to X of the inverse sine it's the same expression except now it is positive