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U1S7V05 Derivative of Sine.txt
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U1S7V05 Derivative of Sine.txt
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#
# File: content-mit-18-01-1x-captions/U1S7V05 Derivative of Sine.txt
#
# Captions for MITx 18.01.1x module [eqD9iHe3g34]
#
# This file has 79 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
So here we go.
Let's check what happens with the sine function.
So I take sine x plus delta x.
I subtract sine x.
And I divide by-- sorry, let's leave a little space there.
Sine x and divide by delta x.
All right, so this is the difference quotient.
And eventually, I'm going I have to take
the limit as delta x goes to 0.
And there's really only one thing
we can do with this to simplify it or change it.
And that is to use the sum formula for the sine function.
So that's this.
That's sine x cosine delta x plus-- Oh,
that's not what it is, huh?
OK.
So what is it?
Sine x sine delta x-- OK, good, you
remember-- plus cosine-- no?
Oh, OK.
So which is it?
OK.
All right, let's take a vote.
Is it sine sine, or is it sine cosine?
OK, so is this going to be-- cosine.
All right.
You better remember these formulas, all right?
OK, it turns out that it's sine cosine cosine sine.
So here we go.
No?
You got to do x here sine delta x.
All right?
So now, there's lots of places to get confused here.
And you're going to need to make sure that you get it right.
All right?
So we're going to put those in parentheses
here sine a plus b is sine a cos b plus cos a sine b.
All right, now, that's what I get over here,
except it was the letter x was a,
and the letter b was delta x.
Now, that's just the first part.
It's just this part of the expression.
I still have to remember the minus sine x.
That comes at the end, minus sine x.
And then I have to remember the denominator, which is delta x.
Okay?
All right, so now, the next thing we're going to do
is we're going to try to group the terms.
And the difficulty with all such arguments is the following one.
Any tricky limit is basically 0/0 when
you set delta x equal to 0.
If I set delta x equal to 0, this is sine x minus sine x.
So it's a 0/0 term.
Here, we have various things which
are zero and various things which are non-zero.
We must group the terms so that a 0 stays over a 0.
Otherwise, we're going to have no hope.
If we get some 1/0 term, we'll get something
meaningless in the limit.
So I claim that the right thing to do here is to notice--
and I'll just point out this one thing-- is that the when delta
x goes to 0, this cosine of 0 is 1.
So it doesn't cancel unless we throw in this extra sine term
here.
So I'm going to use this common factor and combine those terms.
So this is really the only thing that you're
going to have to check in this particular calculation.
So we have the common factor of sine x.
And then that multiplies something that will cancel,
which is cosine delta x minus 1 divided by delta x.
That's the first term.
And now, what's left?
Well, there's a cosine x that factors out.
And then the other factor is sine delta x over delta x.