Derivatives of inverse trigonometric functions
Derivative of inverse cosine
Voiceover: 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 do in this video is to pause it and try to do the same type of proof for the derivative of the inverse cosine of x. 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 set y being equal to this. y is equal the inverse cosine of x, which means the same thing is saying that x is equal to the cosine, cosine of y. I'll just 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 dy, dx, and so we get ... 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 instead of sine y. Well, we know when we saw in the last video from the pythagorean identity 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. I can 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 ... I'm 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 ... I think I lost that color, I'll do it in magenta ... 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 derivative of the inverse sine. Actually, let me put them side by side, and we see that the only difference here is the sign, so let me copy and paste that. I'll 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. A negative 1 over the square root of 1 minus x squared. If we're looking at the derivative with respect to x of the inverse sine, it's the same expression except now it is positive.