Welcome to MySecretMathTutor For this video we are going to work on finding
the derivative of inverse trigonometric functions.
Now these guys are kinda crazy looking, probably have seen them before if you've taken Trigonometry,
but I'm taking about things like inverse tangent, inverse sine or arcsine.
Those are the things we want to take a derivative of.
Now I'm going to jump into an example of these just to give you a sense that these don't
quite look like other derivatives that you are use to.
I'm gong to do a problem like this, y = 3x^2 + cot^-1(x) just to see what we are up against.
Alright, so off I go to take a derivative.
These derivatives really play well the other derivative rules so I'm going to take the
derivative of 3x^2 and then worry about the derivative of inverse cotangent.
So first the derivative of 3x^2, that just gives us 6x.
No problem there.
Now comes the really interesting one, what is the derivative of inverse cotangent.
Well its negative one, all divided by 1 + x squared.
Isn't that crazy?
We could clean this up a little bit, call it 6x minus 1 over 1 plus x squared.
But go ahead and look at that derivative.
Notice how it doesn't have any more trigonometric functions in it.
All of a sudden it just turns into this fraction, and now x squared is on the bottom.
That is not what your intuition says that this probably turns into for derivatives.
So now is a good time to take a look at some of these rules, to see well what is that we
should do when we run across these inverse trigonometric functions.
So there are six functions, and we have six rules.
Don't worry, even though they look kind of complicated, there are only three rules you
really have to remember, and those are the ones for inverse sine, inverse tangent, and
inverse secant.
As soon as you have these rules down, you have them memorized.
You really have the other ones memorized as well.
They are exactly the same, they are usually just different by a negative sign.
And the good way you can connect these two is sine and its cofunction, cosine those are
the ones that are the same, but cosine has the negative part.
Here is inverse tangent and its cofunction, inverse cotangent.
It has the negative sign for inverse cotangent.
So as you can see with these rules, none of them actually contain any trigonometric functions
once you are done taking that derivative.
Whether that is inverse sine or inverse tangent, its derivative is usually a fraction, there
is usually an x squared, or a square root floating around in there.
In a different video I'll explain why these are the correct rules by going over some proofs,
but for now might just want to start memorizing these and getting really familiar with them.
Even though they do look a little bit more complicated than your average derivative rule.
You really do want to get to know these as much as possible so that you can recognize them,
actually for later on when you do integrals.
Alright, so let's just quickly cover these two.
If I take the derivative of inverse sine this is one divided by the square root of one minus
x squared.
If I take the derivative of inverse tangent, then I have one divided by one plus x squared.
And if I have the derivative of inverse secant, then I have one divided by x multiplied by
the square root of x squared minus one.
Now before we jump into two examples I want to really make a note that these definitions
actually may be different than your particular math book.
It all depends on how the math book has set these functions up, how they've defined them.
And you may end up with something slightly different, so yours may actually end up with
some restrictions in there, or may it has absolute values.
Double check your math book before you start on your homework, and you start using these
particular rules.
Alright, let's go ahead and give these guys a try.
See what we have for my examples, and of course, do some derivatives.
Now like I said before these things really play nice with the rest of your derivative
rules, so you still want to know your chain rule, your product rule, quotient rule, all
those things are important, as well as knowing these new derivative rules.
Alright let's go ahead and start off with this guy over here.
So we want to take the derivative of sin^-1 and on the inside it looks like we have a
4x^2.
Since these are a little more complicated, what I like to do when finding their derivatives,
is you know, kinda start off with a template of the what the derivative of inverse sine would look like
So the derivative for the outside is 1 divided by, then we have that square root.
If you look back at my rule sheet it says 1 - x^2.
I'm not going to write that x in there, because that x is really just a stand in for what
ever the inside piece is, and in this case its a 4x^2.
So using that chain rule, the inside would stay exactly the same.
4x^2 Multiplied by the derivative of the inside I have an 8x.
Now too bad, we have a little bit of cleaning up we can do.
Let's go ahead and say that the derivative of y, we'll put that 8x on top, all divided
by the square root of 1 minus, square the four, we get sixteen.
Square the x^2, we get x^4.
Nice.
And there is our derivative.
Notice again, no trigonometric functions, looks a little complicated, but sure enough
that is our derivative.
Alright let's do a very similar one, definitely want to keep in mind the derivative of inverse
sine while we are doing this.
We'll also going to use a chain rule on this one, but notice how this time the inside piece
is the inverse sine.
So let's take care of that outside piece.
We'll bring down the power, reduce it by one.
Now the inside will stay exactly the same, so its just inverse sine of x.
And now we'll multiply this by the derivative of inverse sine.
Here is where we get to use that fraction.
So let's see the derivative of inverse sine, we have, lets see, 1 divided by the square
root of one minus x squared.
Not bad, not bad at all.
So this one of course we have some cleaning up as well.
Might as well combine everything into a single fraction, so we put the 4 and the sine inverse
of x on top.
And we'll just leave all that other square root stuff in the basement.
Alright, so let's do a couple more examples, just so we can get a little bit better at
doing this.
Let's go ahead and throw in a few more of those inverse trigonometric rules, so we can
see what those look like.
Alright.
This next one we'll go ahead and take care of the derivative of e^x multiplied by tan^-1(x).
Now in order to take care of this guy I really want to point out that we are going to have
to use a product rule.
So here we have one function, call this f.
And really this guy over here is a second function, I'll call that g.
Using our product rule then we'll say that it is our first function, just as it is, no
changes multiplied by the derivative of the second function, so I need the derivative
of tan^-1(x).
Ok, we've seen this on our rule sheet.
It's kind of a nice one actually, because this the one that doesn't involve any square
roots, its just 1 divided by 1 + x^2.
Alright, continue on with our product rule.
Now I'll take my second function, unchanged, don't mess with it, don't do anything, multiplied
by the derivative of the first function.
So e^x.
So I just want to point out what I've done here.
So I have my first function, that's just the e^x.
We have the derivative of the second, there is all of that stuff.
Now we have our second function, just as it is and the derivative of the first.
The derivative of a natural exponential of course is just that same natural exponential,
e^x. Ok, so I have everything, looks good.
Now we'll just go ahead and clean it up a bit.
So y' is equal to, let's throw that e^x on top, why not.
1 + x^2 on the bottom, plus I don't know.
I like to put that e^x out front of my inverse tangent, just to make sure everyone knows
that its not on the inside of the inverse tangent.
It's really just being multiplied.
Alright.
On to the next one.
This is y = csc^-1(e^x).
Now I wanted to throw this one in here because if you look at the rules for the derivative
of inverse cosecant, there are actually two little x pieces in there.
So watch for that to show up when we take its derivative.
So the derivative of inverse cosecant.
So negative, divided by, here is where one of those x's goes.
The square root, here is where another one of those x's were, squared minus one.
And that is really just setting up the derivative of the outside.
Now of course we don't have an x, we have something a little more complicated on the
inside, we have e^x.
So that would become the inside here.
So e^x. e^x.
Now we'll multiply by the derivative of the inside.
The derivative of the inside in this case is just e^x. Alright, now there is definitely
some good cleaning up we can do here.
In fact we have a common e^x in the bottom and in the top, so we'll go ahead and get
rid of those.
So we'll say that the derivative of here is a -1 all divided by the square root of e^2x
minus 1.
And now we are done with that derivative.
So these rules they look a little bit complicated.
I highly recommend you start memorizing these as soon as possible, because you will want
to recognize them in reverse when you do get to integrals.
And of course double check your math book to make sure that your definitions match with
what you are going to use on the homework.
Alright, thanks for watching.
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