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For this video we want to start our work with
the basic rules of derivatives.
So essentially we have some sort of function,
and we are curious what is its derivative.
Now in calculus there is tons of different
rules for derivatives and in this video we
are just going to start off with the very
basics, so when you are done with this one,
go ahead and watch my video on say the power
rule, product rule, quotient rule and chain
rule, and you should have a good idea for
derivatives.
Alright, let's go ahead a look some of the
basics that we need to know.
The very first thing that I want to point
out is that I'll often use this little notation,
known as the derivative operator.
Ok.
So what its job is to do, is to say, "I want
to take a derivative of a piece of a function."
So you can think of many other different operators
like here is one, when we were doing the square
root of say a number.
You know.
This says I want to take the square root of
4.
And just like this is an operator, I use it
when I want to take the derivative of something.
So as soon as its done taking its derivative,
then its completely gone.
So as soon as I'm done with taking the square
root of 4, its also completely gone, and I
just write the result.
Ok
Watch for this guy to show up in many different
locations.
Alright, let's see what we have for some of
the basics.
The very first thing you want to know about
derivatives is that you can split them up
over addition and subtraction.
So think of a little situation like this.
Maybe I'm taking the derivative of 5 plus
or minus x.
I can actually take this derivative and apply
it to both of those terms.
That way I'm taking just the derivative of
5 or just the derivative of x.
And this is a really good handy rule when
you have many different pieces to your function.
It allows you to break up that derivative
among all those pieces.
Next is what to do when you have the derivative
of a single constant.
Maybe like the derivative of 7, or this could
be 5, 6 who knows.
When you take the derivative of a single constant
you get 0.
So just 0.
Now keep that in mind that its completely
different if you take the derivative of a
single x, that derivative is actually 1.
When we get to the power rule, you'll see
what we do for higher powers of x, like x^2
and x^3.
But know that a single power you get 1.
Alright.
Now we also have a rule for what to do when
constants are multiplied by something.
So think of 3 multiplied by x^2.
Now in a situation like this, we can actually
move that constant out in front of our derivative.
So then I'm looking at 3 multiplied by the
derivative of everything else.
Now this looks like almost a similar rule
to this one up here, after all they both deal
with constants, but they are different.
Remember in this one we have a constant all
by itself, so its a single constant, and this
one is a constant multiplied by something,
so it moves out front.
Ok.
Alright, and the last of our basics, is a
really neat one.
This is when you take the derivative of a
natural exponential.
So e^x.
When you take its derivative you get the same
thing back.
So it doesn't change it.
So this is a really handy one and its why
you often like to see e^x in a lot of your
calculus problems, because you know that its
derivative is just the same as itself.
Alight.
So let's play around with these basics by
looking at lots of examples and see how it
works ok.
Alight, lets do this first one.
I have the function f(x) = 5 times the square
root of 2
I'm curious, what is the derivative of this
function?
So I want to think, what happens when I take
the derivative of 5 times the square root
of 2.
Now think very carefully about whats going
on here.
This, since there is no x in it what so ever,
is simply a constant.
And of course, according to our basic rules,
the derivative of a constant is 0.
So I can say that the derivative of this function
f(x) is 0.
Just like that.
Alright, let's do something with a few more
things in it.
In this next one, we want to take the derivative
of 3 + x.
so we want to take the derivative of 3 + x.
Ok.
Now notice how this one has two different
pieces to it, and so we want to think of taking
the derivative of each of them.
And we can do it because they are connected
using addition.
So now I want to know, what is the derivative
of 3, and what is the derivative of x.
Well again there are two more derivatives
that come right off of our basic rule sheet.
The derivative of a constant would simply
be 0, and the derivative of a single x, would
be 1.
So I can simply all of this and say what is
the derivative of f?
Well its simply 1.
Alright, not too bad.
Uh, let's look at a few more.
Let's see if we can take the derivative of
5x+1.
So the derivative of f we want the derivative
of 5x+1.
Ok.
Just like before I can see that I have two
pieces that are being combined using addition.
So we'll imagine taking our derivative operator
and putting it one each of those pieces.
So now we want to know, well what is the derivative
of 5x, and what is the derivative of 1.
Now in this first little piece I have a constant
multiplied by x.
So I'm actually going to take this 5 and move
it out in front of my derivative operator.
So 5 times the derivative, of x, plus the
derivative of 1.
And now I have my derivative on these nice
little single components here, and we can
take the derivative of each of those.
So 5 times the derivative of x is 1, plus
the derivative of a constant, 0.
So I know that the derivative for all of f
is simply 5.
Alright, not bad.
This next one has a bunch of different pieces
to it, definitely put your thinking caps on
and see how we go through this one.
So we need the derivative of x - 3x + the
square root of 5.
So even though I have three pieces, they are
all connected using subtraction or addition.
Let take our operator here and put it on all
of those pieces.
So now I"m looking for the derivative of x,
minus the derivative of 3x, and the derivative
of the square root of 5.
Ok, now two of these are not so bad.
The derivative of x is 1, and the derivative
of a single constant is 0.
This one we have a constant multiplied by
an x.
So we are going to move that constant out
in front of its operator.
So the derivative of x is 1.
Minus 3, still need to take the derivative
of x, plus 0.
Ok.
So now this is the last derivative I have
to do.
And of course the derivative of x is 1.
Alight, so what do we really have left?
So the derivative of f(x) is 1 - 3, or a -2
Alright, looking good.
Hopefully its starting to get a little bit
more solid.
Let's look at one last example, and we'll
see how it goes.
This last one I have f(x) = 3e^x - 4x
Alright, so our derivative of f, we need to
figure out the derivative of 3e^x - 4x.
Alright.
So we'll start of by taking our derivative
operator, putting it on each of the pieces.
So now we are looking at the derivative of
the first pieces minus the derivative of the
second piece.
Ok, each of these have a constant multiplied
by something.
So we'll move each of these constants out
in front.
So 3 the derivative of e^x minus 4 the derivative
of x.
Alright, now we can take care of each of these.
The derivative of e^x, that's our special
one, its the same as e^x.
Nothing changes.
The derivative of x is 1, so we have -4 times
1.
Alright let's write down everything we have
left over.
So the derivative of f(x) is 3e^x - 4.
And now you are done.
So you can that as you really take these basic
rules and start to combine them, you can do
the derivatives of some pretty neat functions.
Now of course this isn't all of the rules
of derivatives, again, go ahead and check
out my other videos for the power rule, product
rule, quotient rule, and chain rule.
So you can do even more complicated derivatives.
Alright,
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