Name: Proof of the derivative of sin(x) | Derivatives introduction | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T12:49:48.000Z
Duration: 5 min 52 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

- [Instructor] What we have written here are two

of the most useful derivatives to know in calculus.

If you know that the derivative of sine of x

and actually prove this first derivative.

using the information we're going to do in this one,

that there is a little bit of mathematical

So the derivative with respect to x of sine of x,

by definition, this is going to be the limit

This is really just the slope of the line between the point

Well, we can rewrite sine of x plus delta x

that we learned during our trig identities.

I'll write, rewrite this using our trig identity as cosine,

And then we're going to subtract this sine of x up here

let me see if I can draw a relatively straight line,

So this can be rewritten as being equal to the limit

I have the sum of things up here divided by delta x,

I'm just breaking it up a little bit, plus sine

Well, the limit of a sum is equal to the sum of the limits.

let's see I can rewrite this as cosine of x

of, and let's see I can factor out a sine of x here.

And let's see if I can simplify this even more.

So this left-hand expression I can rewrite.

so we can actually take that outside of the limit.

So we have the cosine of x times the limit

Subtitles ListPlay Video

Proof of the derivative of sin(x) | Derivatives introduction | AP Calculus AB | Khan Academy

entire

expression

straight

negative