00:01
Okay, in this problem, we want to prove that the definition or that the derivative of the cosine of x is negative to sign of x by using the definition of the derivative.
00:12
Okay, and then i wrote an identity we're going to need here.
00:15
So f prime of x equals the limit as h goes to zero, f of x plus h minus f of x over h.
00:29
And you might use delta x instead of h or a doesn't matter okay so that's the limit as h goes to zero so our function is the cosine of x so that's f of x so if you're doing f of x plus h then that's equal to the cosine of x plus h minus f of x which is just the cosine of x all over h okay so now what i'm going to do is in place of the cosine of x plus h i'm going to use this identity where x is a and h is b all right so i get the limit as h goes to zero cosine x times the cosine of h minus the sign of x times the sign of h okay so that's that's what this is equal to minus the cosine of x that's this all over h.
01:45
Okay, now i'm going to rewrite this as two fractions.
01:49
I'm going to put this piece and this piece together, because they both have cosines in them.
01:55
So cosine x, cosine h minus cosine x.
02:02
Well, first, let's just change the order there.
02:10
All right, and then out of the first two, i'm going to factor out a cosine.
02:18
Okay, because we have a cosine x here.
02:21
And a cosine x here...