00:02
Okay, so we have two questions here.
00:06
The first questions is to prove that the derivative of cosine is negative sign of x using the definition of derivative.
00:18
So this is something we just learned when we first studied derivatives that the derivative cosine is negative sign.
00:26
But proving it is a different story.
00:29
So to prove something like this, we're going to use the limit definition of the derivative.
00:33
Derivative so limit as h approach 0 of f of x plus h minus f of x all over h and hopefully what we want will come out which is negative sine x so our f of x is cosine x that's our f of x is cosine x so let's see if we can plug in some stuff here so limit i never write how i write the word limit.
01:11
Okay, cosine of...
01:13
So, cosine of x, or f of x, so f of x plus h is cosine of x plus h minus cosine of x over h.
01:28
So this requires using trick identities for this problem.
01:34
So you can just...
01:36
So let's see what the trick identity says for cosine of...
01:41
A plus b.
01:42
You can't just distribute the cosine because cosine is not a number, it's a function.
01:48
So you can't just think of it like having two times x plus each.
01:52
It doesn't work like that.
01:54
So we got the trick identity.
01:56
Let me write it out here.
02:01
Cosine of a plus b is cosine a, cosine b minus sine a, sine b, minus sine a, sine b.
02:13
So we can treat the a like the x and the b like the h.
02:19
Or you can treat the x like the a and the h like the b.
02:23
So let's see we can apply that identity to our cosine of x plus h.
02:31
So cosine of x plus h is cosine of x times cosine of h minus sine of x times sine of x times sine of h.
02:47
And the identity for sign is similar, but it has plus and it's sine cosine.
02:54
And the identity for cosine is cosine sine.
02:56
Okay, it has a minus.
02:59
All over each.
03:06
Okay, let's see what we can do here.
03:26
Okay, uh, hmm.
03:29
Interesting.
03:31
Oh, i forgot.
03:35
I forgot one thing.
03:36
Okay, i forgot to do, so i did the, cosine of x plus h but i didn't minus out the cosine x which is just cosine x okay all right if you follow the proof in most textbooks what they do next is that they factor they rearrange terms so both of these for example have cosine so cosine of x so let's put them together so so cosine x times cosine h minus cosine x minus sine x minus sine x, sine h, all over h.
04:27
And we can factor our cosine of x for the first two terms.
04:34
So we got cosine of x times cosine of h minus one, all over h.
04:46
We have sine of x times sign of h all over h okay um now using the property of limits properties one of the properties of limits is that if you have a limit of something times something you can rewrite it as the limit of a times the limit of b so if you have the limit of a times b you can rewrite it as limit of a times limit of b so let me do that here so we got limit h approaches 0 of cosine of x times the limit as h approaches 0 of cosine of h minus 1 over h.
05:38
Minus the limit of as x approaches 0 of sine of x over h times the limit as h approaches 0 of sine of x over h times the limit as h approach 0 of sine of h over h away this would just be sign of x because i'm already using this h sorry not that term this h here for the sign of h so if i do another h here would be h square so that would be not right same thing here i just have the h one time not both because then that would be h square let's see we can simplify or use some kind of trick identity or not limit property.
06:39
Okay, so for this one, the limit as h approaches 0 of cosine of x is always going to be cosine of x because the variable is h.
06:55
So you have x here, so that means x is treated like a constant.
06:59
So basically you have a constant function and if you take the limit, it's always going to be the same.
07:04
So that's just going to be cosine of x...