BAM!!!. Mr. Tarrou. In this video we are going to do more examples of chain rule. Only this
time instead of doing just pure algebraic functions, we are going to incorporate some
trig functions. So, we have our six rules for derivation or derivatives of trig functions.
We the derivative of sine is cosine. The derivative of cosine is negative sine. The derivative
of tangent is secant squared. The derivative of cotangent is negative cosecant squared,
The derivative of secant is secant times tangent. And the derivative of cosecant is negative
cosecant times cotangent. However, you might have noticed as I skimmed through those, I
didn't exactly say everything these equations say. Really we have here the derivative with
respect to x of the sine u. That is giving you a hint that there is going to be some
kind of function going on inside here that we have to take care of with the chain rule.
So we are going to have the derivative with respect to x of the sine of u is equal to
the cosine of u. Then we are going to find the derivative of that trig function, but
then because we have another inside function we are going to have to take that chain, extend
it out and find the derivative of that inside function. So we have y equals the tangent
of 3x. Well, this 3x is our u. Ok, that is our inside function. It is the tangent of
something. What is the derivative for tangent? How do you find that? Well, y prime is (the
derivative of y) is equal to, let's see here. The derivative of tangent is secant squared.
It is going to be secant squared of 3x, but you know before... What is the derivative
of tangent x? Secant squared x. But now we have an inside function of 3x. So we are going
to have to find the derivative with respect to x of 3x. And of course are we just learning
this so I am showing you all the steps. A lot of you will just say the derivative of
3x is equal to 3 and you would be absolutely correct. So this becomes y prime is equal
to, I am going to go ahead and move this out front as a coefficient because that is what
we are going to get. The derivative with respect to x of 3x is equal to 3. So we have three
secant squared of three x. Great! So now we have the secant of three x squared minus one.
Same I idea just a different trig function and just a tiny bit more going on inside that
actual function. This might seem more like there actually is a u in there. The 3x, I
mean it is an inside function and we do need to take care of the chain rule. But, I could
definitely see students making a few more mistakes with this kind problem and then here
you can clearly see we have a binomial and there is another function inside of there.
Both of these of course as you can see require the chain rule. So y prime is equal to secant,
now this is our u, so secant of three x squared minus one times tangent of three x squared
minus one. That is our u, so here is our u and we have to find the derivative of u or
the derivative with respect to x of three x squared minus one. Well, the derivative
of three x squared, we are going to apply that power rule and get 6x. This is a constant.
I want to move that 6x out front and get y prime is equal to 6x secant of three x square
minus one times the tangent of three x squared minus one. Let's go on to a couple of more
examples. Or not! I always check my answers as I go, talking, looking around, making sure
I am looking at the camera or whatever. The same thing happens in front of the class.
Sometimes you just think, talk, and write and it does not always quite work out. You
forget where you wanted to go. This may very be an answer that your teacher will accept.
However, sometimes you go in the back of the book and you look up your answers, it would
be nice if they are assigning you the odds so you can check your work, and you do some
work properly here with the second example but it does not look like what is in the back
of the book. You might see this only in terms of sine and cosine. So let's go ahead and
clear this off. In case you need to, secant is one over cosine and tangent is sine over
cosine. So this can be rewritten as y prime is equal to 6x over one times, this is one
over cosine, so one over cosine of three x squared minus one times, tangent is sine over
cosine so the sine of three x squared minus one over cosine of three x squared minus one.
Now ok, I have a 6x on top and sine of three x squared minus one. Then we have cosine times
cosine. Now may not be able to put these together and write cosine squared, except for the fact
that we do have the same angle, the same variable, the same inside function for both of these
trig functions. So they will go together. The final answer, another form of the final
answer, is y prime is equal to 6x times the sine of three x square minus one all over
cosine squared of again 3x^2-1. There would not be much left to the problem unless you
were tempted to cancel these out. Please just in case you are, they do not. These expressions
are inside of math functions. This is inside the sine function and this inside the cosine
function. Now, let't get to the next few examples. So we have our next three examples. We have
y is equal to the sine of four times x to the third. Cosine, I just decided to switch
the trig functions to get you to remember those. y is equal to cosine of 4x to the third.
Then we have cosine cubed, or to the third, of four x. That is not times, is of, that
is a math function. So all of these look similar but they are going to work quite a bit differently
because of where the parenthesis are and where the power is. So we have y equals the sine
of four times x to the third. See, it is just the sine of four. X to the third is not inside
the sine function because of the parenthesis around the sine of four. So this, we are going
to worry about actually calculating that value but that is a constant. So we are going to
use the constant rule for finding derivatives which simply going to be y prime is equal
to the sine of four, that constant value. There is a decimal, there is a fixed value
for that...a fixed answer. This is times the derivative with respect to x of x to the third.
So we are not really doing the chain rule here at all. We are just recognizing that
this is a constant and not getting confused because this is in the section of the chain
rule. We are just going ok, it is a constant, it is the derivative of x to the third power,
so y prime is equal to sine of four times three x squared. And if you would like this
to look a little bit neater you could take the three and move it out front, or actually
the whole three x squared if you want to write that first for some reason. You have three
x squared times the sine of four. 3 times sine of four x squared. What ever you like!
I like this way a little bit better because you won't maybe , especially if you are a
little bit sloppy with your notation, think that the x is inside the sine function. It
is just the sine of four. Actually, my textbooks don't do this very much but I like writing
parenthesis around the angle measure that is in the trig function. That way that little
bit of confusion does not happen. Here we have y equals the cosine of 4x cubed. But
is not the sine function being cubed, it is just the four x. So we are going to go ahead
and apply that power and get y equal to the cosine of, four cubed...four times four is
sixteen times another four is 64 x cubed. Now we are going to find the derivative and
y prime is equal to, the derivative of cosine is negative sine, of 64 x cubed times the
derivative of the inside function. This is a u value here. This isn't just the cosine
of x, it is the cosine of 64 x cubed. We want to find the derivative with respect to x of
64 x cubed. This is just going to be applying the power rule. So we have, let's see here,
we have y prime is equal to, 64 times three... 60 times 3 is 180 and 4 times 3 is 12. I lost
my place there for a second. 60 times 3 is 180 and 4 times 3 is 12, and 180 and 12 is
192. I am going to go ahead and let that 192 float out front. So we are going to have the
negative here and the 192 so negative 192. When we do the power rule we reduce the exponent
by one so that is going to be x squared. That is the derivative of 64 x cubed. Let's just
take the sine of 64 x cubed and stick it at the end. Alright, so let's make sure that
I did not do something silly:D Alright, moving on to our next example. Y equals cosine cubed
of 4x. Now this, this is the way that we write it. But if you are using a calculator, maybe
like a CAS calculator to make sure that your answers are right, or just maybe to help this
problem flow a little bit better, maybe make it look more like chain rule and remind you
of that general power rule, we are going this cosine cubed and move it wrap this entire
trig function in a set of parenthesis and move that three out there. Those are equivalent
statements. So cosine of 4x all cubed. See where the placement and the parenthesis are
different between the second and the third example here. So we are going to have to apply
the chain rule and the general power rule. We are going to take the... Not power rule,
the general power rule, just we are going to apply the chain rule. So we are going to
take the three and move it out front. So we have y prime is equal to three times the cosine
of 4x. I am not finding the derivative of cosine yet, I am using the general power rule
and I am bring this three out front. So it is going to be three times the cosine of 4x
raised to the second power. Now, I need to, this is my inside function. This is my something
cubed. This is my u. I have to find the derivative with respect to x of the cosine of 4x. Ok,
well the derivative of cosine is negative sine, so chain rule. We have y prime is equal
to three times the cosine of 4x. Let's go ahead write that power of two, let's go ahead
and write that down here. So cosine squared of 4x. The derivative with respect to x of
cosine is negative sine of 4x, and again we don't have just a simple x in here, I have
got 4x. I have another inside function. So now I am going to have to apply the chain
rule again. And just because we are learning, the derivative with respect to x of 4x. Ok,
so the derivative with respect to x of 4x is going to be 4, so this is four. This is
all multiplied together, so I am going to let my four float out front and connect with
the three. We get y prime is equal to four times three which is 12 cosine squared of
4x. And what else have I not paid attention to yet? This is a positive three, here we
have a negative one along with that four, so that twelve needs to be negative because
of this negative right here. So negative twelve cosine squared 4x, now we have the negative
out here, sine of 4x. I think we are done with these examples. What else do we have
here. I have got two, I have three more examples involving trig functions. They are rather
long and tedious. I think this lesson is long enough. That is your basic examples for using
chain rule that involves trig functions. If you would like to see three more advanced
examples, I have one here that is going to involve the product rule and the chain rule,
we have many many nested powers of the chain rule and square roots. So, if you need to
see more advanced examples keep watching. If not, I am Mr. Tarrou. BAM!!! Go Do Your
Homework!