- [Instructor] What we're going to do in this video
is try to find the derivative with respect to X
of X squared
sin of X.
All of that to the third power.
And what's going to be interesting is that there's
multiple ways to tackle it.
And I encourage you to pause the video
and see if you could work through it on your own.
So there's actually multiple techniques.
One path is to do the chain rule first.
So I'll just say CR
for chain rule first.
And so I have, I'm taking the derivative with respect to X
of something to the third power.
So, if I take the derivative it would be
the derivative with respect to that something.
So that would be
three times that something
squared
times the derivative with respect to X
of that something.
Where the something, in this case,
is X squared sin of X.
X squared sin
of X.
This is just an application of the chain rule.
Now, the second part, what would that be?
The second part here
do this in another color.
In orange.
Well here, I would apply the product rule.
I have a product of two expressions.
So I would take the derivative of, let me write this down.
So this is gonna be the product rule.
Product rule.
I would take the derivative of the first expression.
So, X, derivative of X squared
is two X.
Let me write a little bit to the right.
This is gonna be two X
times the second expression
sin of X.
Plus the first expression X squared
times the derivative of the second one.
Cosin of X.
That's just the product rule
as applied to this part
right over here.
And all of that, of course, is being multiplied by
this up front.
Which actually, let me just
rewrite that.
So all of this I could rewrite as
let's see, this would be
three times
if I have the product raised to the second power
I could take each of them to the second power
and then take their products.
So X squared squared is
X to the fourth.
And then sin of X squared is
sin
squared of X.
And then all of that is being multiplied by that.
And, if we want, we can algebraically simplify.
We can distribute everything out.
In which case, what would we get?
Well let's see.
Three times two is six.
X to the fourth times X is
X to the fifth.
Sin squared of X times sin of X
is sin of X
to the third power.
And then, let's see, three
so plus three, X to the fourth times X squared is
X to the sixth power.
And then I have sin squared of X
sin squared of X
cosin of X.
So there you have it, that's one strategy
chain rule first, and then product rule.
What would be another strategy
pause the video and try to think of it.
Well, we could just algebraically
use our exponent properties first.
In which case, this is going to be equal to
the derivative with respect to X
of
if I'm taking X squared times sin of X to the third power
instead I could say X to the third power
which is going to be X to the sixth.
And then sin of X to the third power.
Sin of X
to the
third power.
I'm using the same exponent property
that we used right over here to simplify this.
If I have if I'm doing the product things
to some exponent, well that's the same thing
if each of them raised to the exponent
and then the product of the two.
Now how would we tackle this?
Well, I here, I would do
the product rule first.
So let's do that.
So let's do the product
rule.
So, we're gonna take
the derivative of the first expression.
So, derivative of X to the sixth
is six
X to the fifth.
Times the second expression.
Sin
to the third of X.
Or, sin of X to the third power.
Plus the first X to the sixth
times the derivative of the second
and I'm just gonna write that
D DX of
sin of X
to the third power.
Now, to evaluate this right over here
it does definitely make sense
to use the chain rule.
Use
the chain rule.
And so what is this going to be?
Well I have the derivative of something
to the third power.
So that's going to be three
times that something squared
times the derivative of that something.
So in this case, the something is sin of X.
And the derivative of sin of X is cosin of X.
Then I have all of this business over here.
I have six X to the fifth.
Sin to the third
or sin of X to the third power.
Plus
X to the sixth.
And if I were to just simplify this a little bit
in fact, you see it very clearly.
These two things are equivalent.
This
this term is exactly equivalent to this term
the way it's written.
And then this is exactly,
if you multiply three times X to the sixth
sin of X squared
cosin of X.
So, the nice thing about math
if we're doing things that make logical sense
we should get to the same endpoint.
But the point here is that there's multiple strategies.
You could use a chain rule first
and then the product rule.
Or you could use a product rule first,
and then the chain rule.
In this case, you could debate which one is faster.
It looks like the one on the right
might be a little bit faster.
But
sometimes
these two are pretty close.
But sometimes it'll be more clear than not
which one is preferable.
You really want to minimize the amount of hairiness,
the amount the number of steps
the chances for careless mistakes you might have.