Chain Rule Example 3 and Product Rule Example 5

Hi everyone!
Welcome back to integralCALC.com.
We're going to be doing a chain rule problem today
with the trigonometric function here,
y equals sine cube x
sine cubed of x times tangent to the 4x
or sorry, times tangent of 4x.
I've written the chain rule, or sorry,the product rule formula up here
because we'll need it to complete this problem since we have
the multiplication of two different terms:
one being the sin^3(x) and the other being tan(4x).
So, the first thing that I want to do is
just change the way this problem is written to make it easier on ourselves. I think it's easier...
I would like to move the
the exponent here, cubed, out to the outside here
because it's easier for me to visualize.
The important thing and the reason that it's not written this way
is because a lot of people tend to write sin(x^3),
which is not the same. Here, the cubed is on the sin, and here, is on the x.
It's very important to write the parenthesis here because that means that the whole thing is cubed
versus just sin(x^3).
But I think it will help us with chain rule to visualize because this throws a lot of people off when it's in the middle.
So, I'm just going to rewrite it that way, making sure to add the parenthesis and then we'll take it off later.
So, rewrite the problem, and then the next thing we need to do for product rule,
for the purpose of product rule, is assign f(x) and g(x) to terms in our problem.
So, it doesn't matter which one. I'm going to go ahead and call f(x)
sin(x)...
(sin x)^3
and we'll call g(x)
tan(4x).
And this will allow us to keep things straight as we
write out what we need to hear from product rule.
So, this is...
this is the way that we apply product rule when we're taking the derivative.
So let's go ahead and write out the derivative y'.
We'll set it equal to... First, we need f(x) which we...
and if we just leave it along, we don't need to take the derivative,
so, (sin x)^3
(sin x)^3
times the derivative of g(x).
So in this case, I'm just going to go ahead and write out the whole problem and then we'll take the derivative later.
So, that's going to be times the derivative of tan(4x),
then, we have to write g(x) there,
leaving it untouched, so tan(4x),
and then multiply that by the derivative of f(x),
so, I'll go ahead and just write out derivative, (d/dx),
of (sin x)^3.
Okay, so, this is as applying product rule.
Now we can actually go ahead and start taking the derivative
and it routed out at this because we're done with product rule now,
and now we can just think about chain rule.
And remember, with chain rule, that we're always working from the outside in,
taking the derivative one step at a time, first with the
the outside part of the function and then working our way in step by step.
So, when we have y' here, so y' equals...
We can leave the (sin x)^3,
and I'll go ahead and rewrite it the way it supposed to be
here first, so sin^3(x).
Then the derivative of tan(4x), will the derivative of tan(x),
is sec^2(x).
is
So, we can go ahead and write sec^2,
because we have 4x it will be sec^2(4x).
But, that only addresses the
the tangent part here; remember we're working from the outside in.
So we took the derivative, talking about tangent
that was the outside. But now we're moving even further inside the function.
We still have to multiply by the derivative of 4x, we can't
we can't just leave it this way.
So, we'll go ahead and multiply
by the derivative of 4x.
Again, here, working from the outside in,
let's say plus tan(4x),
and then taking the derivative of (sin x)^3.
Again, we will start with the outside. So because we have the cubed exponent out here,
we've bring that out in front, right? taking the derivative using the power rule,
we multiply by that 3 in front,
3 times (sin x),
and we subtract 1 from the exponent
so we get... sorry that's tiny
so we get a 2 here. However,
we worked from the outside and we did the outer part
but we still have to multiply by the derivative of the inside, (sin x),
we can't just leave that along. So, let me go ahead and write out the
multiplied by the derivative, ((d/dx)(sin x)).
So, I hope
I hope you can see how we've been working from the outside in.
let me go ahead and
Let me go ahead and
erase all this stuff that we don't need any more.
So, working from the outside in,
we still need to multiply by the derivative of 4x and by
the derivative of (sin x).
So let's go ahead and rewrite this up here,
y' equals
sin^3(x)
times (sec^2(4x),
and then the derivative of 4x is simply 4,
so we'll go ahead and multiply by 4,
and we're done with that because there's nothing
further inside of the function that we were able to take the derivative of 4x by itself and get 4.
So we're done with that and then we add to that tan(4x),
times.. let's see
3, and we'll change this to
sin^2(x), which is the proper way to write it,
and then the derivative of sin(x) is cos(x).
And technically we would need to
multiply by the derivative of x
alone because that would be properly applying chain rule.
Working from the outside,we would just do sin,
then we will multiply by the derivative of x because it's the inside.
But the derivative of x is just 1, so, we actually don't need to
to multiply by anything, it would just being redundant. So,
we've gone ahead and taken the derivative of everything and the only thing we need to do now
is just simplify. So, I'll move this ...
the 4 out in front as the coefficient
and I'll also be moving the 3 out in front, and reordering.
So y' equals 4sin^3(x)sec^2(4x) plus...
And then here we have sin, cos, and tan.
That would be the proper order to write those trigonometric identities.
You would always want to write them
first sin, then cos, then tan, so, I'm going to do sin^2(x) first,
cos(x) second,
and tan(4x) third,
and that's our final answer.
So, hope that helped and I'll see you guys next time.
Bye!!!