Derivatives of Inverse Functions and Inverse Trig
October 2, 2026

[00:00:05] Okay. Tell me something, anything you know about inverse functions.
[00:00:16] Hold on. That's right. Yeah.
[00:00:17] Like the coordinates that look like flipped like that. So if a comma b is in F, then what?
[00:00:36] B comma a is in F inverse. That's literally the definition is you just flip all the ordered
[00:00:46] pairs. Somebody else tell me something.
[00:01:05] If F is a one to one function,
[00:01:18] then F inverse exists. Okay. So we're guaranteed to have inverse.
[00:01:34] If it is a one to one function, then what is a one to one function?
[00:01:41] Use the vertical and horizontal line tests.
[00:01:49] Every X corresponds only one Y. That makes it a function. And if every Y corresponds only one X,
[00:01:57] then it's a one to one function. Okay. Two Y values never get repeated. Okay. Somebody else.
[00:02:10] This is very, very important. F composed with F inverse
[00:02:20] gets you right back where we started and vice versa.
[00:02:30] I think we talked about that the other day. Right. E to the X and log base E of X or inverse.
[00:02:40] Because of that, we can plug them in. This equals X and LN of E to the X equals X.
[00:02:50] Very, very important. We're going to use that today. What else?
[00:03:07] Nothing else. How are their graphs related?
[00:03:20] We literally switch all of the X and Y coordinates and it gets reflected about the line.
[00:03:29] Y equals X.
[00:03:39] See if I can do this.
[00:03:49] Something like that.
[00:03:53] How are the domain and ranges related?
[00:04:03] Yeah. Opposites are flipped. It has everything to do with this, right?
[00:04:06] If b is in the range of f, then b is in the domain of f inverse. So the domain
[00:04:18] of F is the range of F inverse and the range of F is the domain of F inverse.
[00:04:42] That's pretty much it.
[00:05:02] So let's say we have some function, F of X and its inverse is F inverse of X.
[00:05:12] Our goal is to figure out F inverse prime, the derivative.
[00:05:24] How are we going to do this?
[00:05:32] We don't even know what the functions are.
[00:05:42] We're going to start with an equation that we know.
[00:05:47] We're going to start with one of these two equations.
[00:05:52] And we're going to differentiate it using what kind of differentiation?
[00:05:57] Implicit because we got functions within functions.
[00:06:02] Or we got functions on both sides.
[00:06:05] Are we going to use this one or are we going to use this one?
[00:06:09] Any thoughts?
[00:06:13] One will work. One's not going to work.
[00:06:17] Why a top one?
[00:06:26] Close. Let's start with the top one and then see what happens.
[00:06:42] You'll see why in a second.
[00:06:45] So we use implicit to differentiate the left-hand side.
[00:06:49] And since we have a function inside a function, what rule do we have to use?
[00:06:54] Chain rule.
[00:06:56] So we got F prime of the inside
[00:07:04] times the derivative of the inside, which is F inverse prime.
[00:07:16] And then the derivative of X is with respect to X, which is one or DX DX.
[00:07:27] This is what we're looking for right here.
[00:07:29] The derivative of F inverse.
[00:07:31] That's why we chose the top function because we multiplied by the derivative of this.
[00:07:37] It gives us this.
[00:07:38] If we were to switch it, it wouldn't help us.
[00:07:42] And then we simply divide by that.
[00:08:02] OK, this is the derivative of the inverse of a function.
[00:08:08] Even if you don't know what the function is, if we know the derivative of our function F,
[00:08:15] we plug into it the inverse.
[00:08:19] This is it.
[00:08:20] So let's do an example.
[00:08:27] We'll give you two functions that are inverses.
[00:08:40] What's that?
[00:08:43] I like that.
[00:08:48] Let's go with e to the x.
[00:08:51] We know the inverse is ln of X.
[00:08:57] We know the derivative of the inverse, right?
[00:09:01] We just did that last class.
[00:09:03] The derivative is what?
[00:09:05] One over X.
[00:09:07] But let's show it using that.
[00:09:12] That rule, F inverse.
[00:09:16] The derivative is one over F prime composed with the inverse.
[00:09:33] What is F prime of X?
[00:09:36] E to the X.
[00:09:38] So this is going to be one over E to the inverse.
[00:09:47] And the inverse is ln of X.
[00:09:52] And what is E to the ln of X?
[00:09:57] Just X.
[00:09:58] Just X.
[00:09:59] There it is.
[00:10:00] We got it.
[00:10:02] Even if we didn't know the derivative here.
[00:10:07] Let's try another one.
[00:10:09] How about X cubed?
[00:10:18] What's the inverse of X cubed?
[00:10:23] The cube root.
[00:10:26] Let's write it like this.
[00:10:27] One third power.
[00:10:28] That is the cube root.
[00:10:33] Obviously, we don't need that little rule.
[00:10:36] We know the derivative of this.
[00:10:42] One third x to the negative two-thirds power.
[00:10:47] Negative two thirds.
[00:10:56] But let's show it.
[00:10:59] F inverse prime is one over F prime.
[00:11:06] Of.
[00:11:12] Compose with F inverse.
[00:11:24] Deriv X cubed is three X squared.
[00:11:30] So it's one over.
[00:11:34] Three F inverse squared.
[00:11:41] Which is one over three times the cube root of X.
[00:11:48] Squared.
[00:11:50] Which is the same thing as.
[00:11:59] This makes sense.
[00:12:00] Okay, let's do some other examples.
[00:12:08] So we have no idea what the equation for F inverse is.
[00:12:12] We just give it a what six order pairs.
[00:12:18] Okay, we can find F inverse.
[00:12:25] Sorry, I screwed this up.
[00:12:28] This should be the derivative of F not the inverse of F.
[00:12:42] Okay, I want to know F inverse prime.
[00:12:48] Of two.
[00:12:56] No, not up to.
[00:13:16] Hold on, let me take for a sec.
[00:13:23] Oh, yes, that is right.
[00:13:26] How long were we going to do this?
[00:13:31] F inverse prime of X is one over F prime of inverse.
[00:13:41] X.
[00:13:53] So by that is one over.
[00:13:56] F prime of F inverse.
[00:14:01] Of two.
[00:14:14] Now we got to figure out this number.
[00:14:19] I don't have an F inverse column.
[00:14:27] How do I figure this out?
[00:14:34] Okay, we do know three.
[00:14:48] It's not written down.
[00:14:50] We do know three order pairs of F inverse.
[00:14:51] What's three order pairs?
[00:14:55] Seven, five; three, two; and eight, three.
[00:15:14] Okay, we don't know this.
[00:15:16] Sorry, I did screw this up.
[00:15:19] Let's call this three.
[00:15:24] Sorry.
[00:15:29] We don't know what f inverse of two is.
[00:15:31] How do we get these?
[00:15:35] If five comma seven is in F, then seven comma five is in F inverse.
[00:15:41] Okay, now we know F inverse of three, which is two.
[00:15:56] And what is F prime of two?
[00:16:04] That's it.
[00:16:21] Okay, so we know the slope of the tangent line of F inverse at X equals three.
[00:16:25] We have no idea what the function looks like, but we do not need it because we have this formula.
[00:16:45] Okay, what's the only functions that we don't know the derivative of yet?
[00:16:56] Inverse trig functions.
[00:16:58] We know the derivative of all other functions.
[00:17:01] Polynomial, rational, radicals, exponential, logarithm.
[00:17:05] We know the derivative of all six trig functions,
[00:17:07] but we haven't talked about the inverse of.
[00:17:11] The derivative of.
[00:17:16] So.
[00:17:27] What is the inverse of sine?
[00:17:33] We call this arcsine.
[00:17:37] Or we call it sine inverse.
[00:17:42] Okay.
[00:17:45] A quick review before we actually do this.
[00:17:54] This is sine, right?
[00:17:58] How do we find the inverses?
[00:18:03] The inverse relation is x equals sine of y.
[00:18:12] We just switch the order pair.
[00:18:15] And this function is our inverse.
[00:18:18] Okay, and then we graph it.
[00:18:25] It looks like.
[00:18:40] So, my God, except that's not a function.
[00:18:43] So what do we do?
[00:18:49] Because sine is not a one-to-one function, it does not have an inverse over all real numbers.
[00:18:57] But wherever it's one to one, it does have an inverse.
[00:19:02] So here's where we cut it off.
[00:19:06] We're going to go from here.
[00:19:10] This is going to be the domain.
[00:19:14] Okay, which is negative pi over two to pi over two.
[00:19:26] So we have to cut this off.
[00:19:35] It looks something like that.
[00:19:37] It's a very small function.
[00:19:40] The domain of this is the domain.
[00:19:47] We only plug in angles from negative pi over two to pi over two.
[00:20:02] No, I take this back.
[00:20:12] Let's go back to time.
[00:20:14] What is the domain of sine?
[00:20:23] The domain of sine is all real numbers.
[00:20:27] What is the range?
[00:20:33] From negative one to one.
[00:20:35] Sorry, I screwed this up.
[00:20:38] Actually, that was correct.
[00:20:40] Except I screwed up the.
[00:20:48] I screwed up the yellow part.
[00:20:51] We'll come back in a second.
[00:20:52] The range of sine becomes the domain of sine inverse.
[00:21:01] In arcsine, or sine inverse, we input a value
[00:21:06] right, as a y coordinate from negative one to one, and it outputs an angle.
[00:21:12] Some angle between negative pi over two to pi over two.
[00:21:19] Okay, so this is pi over two.
[00:21:24] This is negative pi over two.
[00:21:28] We cut it off right here at negative one and at one.
[00:21:47] Okay, that was just a review of what sign and arc sign is.
[00:21:52] Now let's find the derivative of it.
[00:21:56] We don't know it, so we can't do anything with this.
[00:22:00] But what do we know?
[00:22:03] Its derivative is one over f prime evaluated at f inverse of x.
[00:22:19] F prime evaluated at f inverse of x.
[00:22:27] Here, f of x is sine.
[00:22:32] F inverse is arcsine.
[00:22:42] What is the derivative of sine?
[00:22:45] Cosine.
[00:22:48] This is one over cosine of the inverse, which is arcsine x.
[00:23:03] Okay, this is the derivative.
[00:23:08] What do we call this circle?
[00:23:16] Uh-oh.
[00:23:17] Circle radius one is called.
[00:23:22] Oh boy, people.
[00:23:30] Let's try this one more time.
[00:23:31] Somebody tell me, what do we call a circle radius one?
[00:23:36] The unit circle.
[00:23:37] Thank you.
[00:23:38] Well done.
[00:23:43] Circle with the radius is one unit, whatever that might be.
[00:23:46] One point.
[00:23:49] Called the unit circle.
[00:24:07] Let's call this X.
[00:24:13] Let's call this Y.
[00:24:24] A and B.
[00:24:43] And let's call this theta.
[00:24:58] Okay, I changed it because this X was not the same thing as the other X.
[00:25:03] When you plug a number between negative one and positive one,
[00:25:12] it's this, it's this length here.
[00:25:18] Why is that?
[00:25:25] Arcsine gives you an angle
[00:25:31] such that the sine of the angle is this length.
[00:25:37] Right, what is sine of theta here?
[00:25:43] It's the, it's this value, B.
[00:25:46] We agree?
[00:25:47] Okay.
[00:25:57] So this is my B value.
[00:26:03] Okay, and this outputs an angle.
[00:26:16] This gives us our theta.
[00:26:19] This one right here.
[00:26:21] You with me so far?
[00:26:29] And what is cosine of that angle?
[00:26:36] It is A.
[00:26:38] Okay.
[00:26:42] Arcsine of b is the angle.
[00:26:45] This inner part is the angle.
[00:26:51] We're almost there.
[00:26:59] This was our input right here.
[00:27:01] It wasn't really X, it was B.
[00:27:03] Let's call B for a second.
[00:27:07] The derivative is one over A.
[00:27:11] But we can put that in terms of B.
[00:27:15] I should have called that X, but I screwed up.
[00:27:19] Okay, what is this in terms of our number B?
[00:27:29] What's this length right here?
[00:27:33] Because we're on the unit circle, it's of length one.
[00:27:36] Thank you.
[00:27:38] Okay, the same magic people.
[00:27:43] And then we can set up the Pythagorean theorem.
[00:27:46] Yeah?
[00:27:47] A squared plus B squared equals one squared.
[00:27:50] We agree?
[00:27:52] And if we solve for A here, A squared is one minus B squared.
[00:28:01] And then A will either equal the positive or the negative.
[00:28:08] Square root of one minus B squared.
[00:28:11] Which one is it?
[00:28:12] Is it a positive or negative?
[00:28:15] Why positive?
[00:28:18] We're in the first quadrant.
[00:28:23] And even if our angle is down here,
[00:28:28] which would be, no, it would be this angle.
[00:28:32] Okay, the A value is still positive.
[00:28:35] Because cosine of any angle in quadrant four is still the positive A.
[00:28:43] Because all students take calculus.
[00:28:48] So long story short,
[00:28:50] this is the square root of one minus B squared.
[00:28:57] Which really was our input here.
[00:29:01] I should have put a B here.
[00:29:05] Okay, so that was the proof that the derivative, that was the proof,
[00:29:12] of arc sine.
[00:29:16] This inverse trig function turns out to be this algebraic function,
[00:29:21] one over the square root of one minus x squared.
[00:29:32] It's kind of cool.
[00:29:34] It's kind of weird.
[00:29:35] The derivative of some trig function, well it's an inverse trig,
[00:29:39] turns out to be this radical function.
[00:29:43] But there it is.
[00:29:46] Let's do the same thing with arccosine.
[00:29:53] Let's call it f.
[00:30:09] So
[00:30:25] we're going to have to memorize all of them.
[00:30:34] We never proved it.
[00:30:36] Now you guys know it.
[00:30:49] If you ever forget, just prove it.
[00:30:52] Exactly, I like this.
[00:30:59] Okay, this one's going to work out with x's.
[00:31:01] It'll be much better.
[00:31:03] Okay, so this is one over.
[00:31:08] F prime is negative sine, evaluated at the inverse, arccosine x.
[00:31:22] Again, this is the derivative.
[00:31:24] We could stop here, but it will, turns out it simplifies.
[00:31:29] Let's see if we can do this.
[00:31:33] Correctly this time.
[00:31:54] Okay, now x will work out quite nicely.
[00:32:14] So, again, we're inputting some length between negative one and positive one.
[00:32:23] So arc cosine.
[00:32:31] We're inputting this length x.
[00:32:36] And it outputs an angle.
[00:32:38] It outputs this angle, theta.
[00:32:50] And what is sine of this angle?
[00:32:57] Y.
[00:33:02] Okay, and then the last thing we need to do is change this in terms of x.
[00:33:12] What's the hypotenuse?
[00:33:14] One.
[00:33:14] One, thank you.
[00:33:16] Because it's the unit circle.
[00:33:19] So we got x squared plus y squared equals one squared.
[00:33:24] Y squared equals one minus x squared.
[00:33:29] Y equals plus or minus the square root of one minus x squared.
[00:33:46] Is it the plus or the minus?
[00:33:54] Y plus.
[00:33:56] Because we're in quadrant one.
[00:34:00] If we were in this quadrant, okay, because the range of arc cosine is between zero and pi.
[00:34:15] Okay, this would be my x, this would be my y.
[00:34:21] The y is still positive in quadrant two, we agree?
[00:34:24] So it has to be positive.
[00:34:27] So we can get rid of this.
[00:34:30] And so it is the square root of one minus x squared.
[00:34:39] Okay, so long story short, the derivative of arc cosine,
[00:34:49] We normally also call it cosine inverse, but it's nice to distinguish it here.
[00:34:56] is simply negative one over the square root of one minus x squared.
[00:35:03] Which is the same as derivative of sine inverse, except it's negative.
[00:35:15] So should we do tangents?
[00:35:24] Let's do tangents for fun.
[00:35:27] For fun. Definitely fun.
[00:35:50] I'm using a different letter because it won't work out that nice.
[00:36:05] Arc tan, same thing as tan inverse.
[00:36:09] A, A, A, A. It doesn't matter. Let's go A. Why not?
[00:36:27] Sorry, what is the derivative of tan?
[00:36:29] Secant squared.
[00:36:31] Secant squared.
[00:36:33] Let me write it like that, secant squared.
[00:36:42] F inverse prime is one over F prime composed with the inverse.
[00:36:54] This is going to be one over secant squared composed with tan inverse.
[00:37:19] We proved that?
[00:37:22] I do.
[00:37:22] Okay.
[00:37:38] Uh oh.
[00:38:07] Go and look back.
[00:38:10] We are not plugging in just a y value or just an x value; we are plugging in y over x.
[00:38:18] the ratio y over x okay wherever it is and this outputs an angle between
[00:38:31] negative pi over 2 pi over 2 okay and that angle would be theta whatever it is
[00:38:44] and what is secant of that angle secant is 1 over cosine okay so 1 over
[00:39:43] a was we agree it's y over x okay so we need to put in terms of a not just x
[00:40:17] okay so a is y over x so x is y over a
[00:40:54] X squared plus y squared is one. We get
[00:41:03] that by the Pythagorean theorem x squared equals 1 minus y squared
[00:41:22] X equals the square root of one minus y squared. Why is it guaranteed to be
[00:41:38] be positive actually it might not be guaranteed in the first quadrant is but I guess my we
[00:42:27] didn't have to have an angle in the first one it could have been down here but it doesn't matter
[00:43:07] give me one more sec we're almost done
[00:43:13] if I substitute in a here okay which
[00:43:26] becomes x squared plus a squared x squared equals 1 we agree which is 1 plus a squared
[00:43:43] times x squared so x squared is 1 over 1 plus a squared so x is either plus or minus the square
[00:44:03] root of 1 over let's do this 1 plus a squared okay this is where sorry took a long time this
[00:44:15] is what we're gonna plug in for x right here so we got 1 over 1 over where is it the square root
[00:44:37] what is 1 over 1 over it's just becomes this in the numerator it took a long time and then
[00:44:55] that simplifies to one over one plus a squared. That was messy because
[00:45:15] it's a ratio it's not just the x to the y coordinate so what is the derivative of arc tan
[00:45:26] or tan inverse is simply 1 over 1 plus x squared which is a nice little rational function okay
[00:45:42] okay these three are the only three we need for this class I don't know why the the blue
[00:45:49] book we use they like to put in use wait this is just saying if there's another function inside
[00:46:12] it's not just x it's 1 over it times the derivative what our u is okay it's just putting the chain
[00:46:24] rule in there if there's something else inside of it okay so here are the other three they're messy
[00:46:36] the other three involve these absolute values, and we don't need them this time. We need to memorize the three:
[00:46:43] arcsine arc cosine and arc tangent we wrote those three down already yeah okay I think we're gonna
[00:46:55] stop there