18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 3
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Professor: In today's lecture I want to develop several more formulas that will allow
us to reach our goal of differentiating everything. So these are derivative formulas,
and they come in two flavors. The first kind is specific, so some specific function
we're giving the derivative of. And that would be, for example, x^n or (1/x) . Those
are the ones that we did a couple of lectures ago. And then there are general
formulas, and the general ones don't actually give you a formula for a specific
function but tell you something like, if you take two functions and add them
together, their derivative is the sum of the derivatives. Or if you multiply by a
constant, for example, so (cu), the derivative of that is (cu)' where c is constant.
All right, so these kinds of formulas are very useful, both the specific and the general
kind. For example, we need both kinds for polynomials. And more generally, pretty
much any set of forumulas that we give you, will give you a few functions to start
out with and then you'll be able to generate lots more by these general formulas. So
today, we wanna concentrate on the trig functions, and so we'll start out with some
specific formulas. And they're going to be the formulas for the derivative of the sine
function and the cosine function.
So that's what we'll spend the first part of the lecture on, and at the same time I
hope to get you very used to dealing with trig functions, although that's something
that you should think of as a gradual process.
Alright, so in order to calculate these, I'm gonna start over here and just start the
calculation. So here we go. Let's check what happens with the sine function. So, I
take sin (x delta x), I subtract sin x and I divide by delta x. Right, so this is the
difference quotient and eventually I'm gonna 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 or change it, and
that is to use the sum formula for the sine function. So, that's this. That's sin x co
delta x plus
Oh, that's not what it is? OK, so what is it? Sin x sin delta x. OK, good. Plus cosine.