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Product rule proof | Taking derivatives | Differential Calculus | Khan Academy - YouTube
Channel: Khan Academy
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- [Voiceover] What I
hope to do in this video
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is give you a satisfying
proof of the product rule.
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So let's just start with our
definition of a derivative.
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So if I have the function F of X,
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and if I wanted to take
the derivative of it,
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by definition, by definition,
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the derivative of F of X is the limit
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as H approaches zero,
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of F of X plus H minus F of X,
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all of that over, all of that over H.
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If we want to think of it visually,
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this is the slope of the
tangent line and all of that,
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but now I want to do something
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a little bit more interesting.
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I want to find the derivative
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with respect to X, not just of F of X,
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but the product of two functions,
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F of X times G of X.
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And if I can come up with
a simple thing for this,
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that essentially is the product rule.
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Well if we just apply the
definition of a derivative,
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that means I'm gonna take the limit
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as H approaches zero,
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and the denominator I'm gonna have at H,
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and the denominator,
I'm gonna write a big,
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it's gonna be a big rational expression,
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in the denominator I'm gonna have an H.
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And then I'm gonna evaluate
this thing at X plus H.
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So that's going to be F of X plus H,
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G of X plus H and from
that I'm gonna subtract
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this thing evaluated F of X.
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Or, sorry, this thing evaluated X.
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So that's gonna be F of X times G of X.
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And I'm gonna put a
big, awkward space here
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and you're gonna see why in a second.
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So if I just, if I evaluate this at X,
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this is gonna be minus
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F of X, G of X.
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All I did so far is I just
applied the definition
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of the derivative, instead
of applying it to F of X,
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I applied it to F of X times G of X.
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So you have F of X plus H, G of X plus H
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minus F of X, G of X,
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all of that over H.
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Limit as H approaches zero.
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Now why did I put this
big, awkward space here?
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Because just the way I've
written it write now,
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it doesn't seem easy to
algebraically manipulate.
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I don't know how to evaluate this limit,
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there doesn't seem to be
anything obvious to do.
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And what I'm about to show you,
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I guess you could view it
as a little bit of a trick.
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I can't claim that I would
have figured it out on my own.
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Maybe eventually if I
were spending hours on it.
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I'm assuming somebody was
fumbling with it long enough
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that said, "Oh wait, wait.
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"Look, if I just add and
subtract at the same term here,
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"I can begin to
algebraically manipulate it
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"and get it to what we all know
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"as the classic product rule."
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So what do I add and subtract here?
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Well let me give you a clue.
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So if we have plus,
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actually, let me change this,
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minus F of X plus H, G of X,
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I can't just subtract, if I subtract it
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I've got to add it too, so
I don't change the value
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of this expression.
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So plus F of X plus H, G of X.
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Now I haven't changed the value,
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I just added and
subtracted the same thing,
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but now this thing can be manipulated
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in interesting algebraic ways to get us to
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what we all love about the product rule.
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And at any point you get inspired,
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I encourage you to pause this video.
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Well to keep going, let's just keep
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exploring this expression.
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So all of this is going to be equal to,
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it's all going to be equal to the limit
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as H approaches zero.
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So the first thing I'm gonna do is
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I'm gonna look at, I'm
gonna look at this part,
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this part of the expression.
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And in particular,
let's see, I am going to
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factor out an F of X plus H.
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So if you factor out an F of X plus H,
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this part right over here is going to be
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F of X plus H,
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F of X plus H,
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times
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you're going to be left
with G of X plus H.
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G of, that's a slightly
different shade of green,
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G of X plus H, that's that there,
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minus G of X,
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minus G of X,
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oops, I forgot the parentheses.
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Oops, it's a different color.
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I got a new software program
and it's making it hard
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for me to change colors.
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My apologies, this is not
a straightforward proof
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and the least I could do is
change colors more smoothly.
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Alright, (laughing) G of X plus H
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minus G of X, that's that
one right over there,
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and then all of that over this H.
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All of that over H.
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So that's this part here
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and then this part over here
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this part over here, and
actually it's still over H,
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so let me actually circle it like this.
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So this part over here
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I can write as.
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So then we're going to have plus...
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actually here let me,
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let me factor out a G of X here.
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So plus G of X
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plus G of X times this F of X plus H.
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Times F of X plus H
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minus this F of X.
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Minus that F of X.
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All of that over H.
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All of that over H.
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Now we know from our limit properties,
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the limit of all of this business,
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well that's just going
to be the same thing
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as the limit of this as H approaches zero
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plus the limit of this
as H approaches zero.
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And then the limit of the product
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is going to be the same thing
as the product of the limits.
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So if I used both of
those limit properties,
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I can rewrite this whole
thing as the limit,
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let me give myself some real estate,
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the limit as H approaches
zero of F of X plus H,
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of F of X plus H times,
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times the limit
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as H approaches zero,
of all of this business,
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G of X plus H minus G of X,
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minus G of X,
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all of that over H,
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I think you might see where this is going.
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Very exciting.
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Plus,
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plus the limit,
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let me write that a
little bit more clearly.
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Plus the limit as H
approaches zero of G of X,
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our nice brown colored G of X,
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times, now that we have our product here,
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the limit,
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the limit
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as H approaches zero of F of X plus H.
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Of F of X plus H minus F of X,
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minus F of X,
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all of that,
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all of that over H.
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And let me put the parentheses
where they're appropriate.
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So that,
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that,
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that,
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that.
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And all I did here, the limit,
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the limit of this sum,
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that's gonna be the sum of the limits,
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that's gonna be the limit of this
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plus the limit of that,
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and then the limit of
the products is gonna be
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the same thing as the
product of the limits.
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So I just used those
limit properties here.
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But now let's evaluate them.
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What's the limit,
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and I'll do them in different colors,
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what's this thing right over here?
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The limit is H approaches
zero of F of X plus H.
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Well that's just going to be
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F of X.
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Now, this is the exciting part,
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what is this?
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The limit is H approaches
zero of G of X plus H
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minus G of X over H.
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Well that's just our,
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that's the definition of our derivative.
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That's the derivative of G.
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So this is going to be,
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this is going to be the
derivative of G of X,
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which is going to be G prime of X.
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G prime of X.
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So you're multiplying these two
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and then you're going to have plus,
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what's the limit of H
approaches zero of G of X?
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Well there's not even any H in here,
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so this is just going to be G of X.
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So plus G of X
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times the limit,
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so let's see, this one is in brown,
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and the last one I'll do in yellow.
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Times the limit as H approaches zero,
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and we're getting very close,
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the drum roll should be starting,
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limit is H approaches zero of F of X
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plus H minus F of X over H.
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Well that's the definition
of the derivative
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of F of X.
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This is F prime of X.
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Times F prime of X.
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So there you have it.
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The derivative of F of
X times G of X is this.
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And if I wanted to write
it a little bit more
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condensed form, it is equal to,
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it is equal to F of X
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times the derivative
of G with respect to X
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times the derivative
of G with respect to X
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plus G of X,
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plus G of X times the derivative
of F with respect to X.
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F with respect to X.
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Or another way to think about it,
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this is the first function
times the derivative
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of the second plus the second function
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times the derivative of the first.
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This is the proof, or a proof,
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there's actually others
of the product rule.
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