Name: Proof of the derivative of cos(x) | Derivative rules | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T12:49:56.000Z
Duration: 3 min 18 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 I wanna do in this video

is make a visual argument as to why the derivative with

respect to x of cosine of x is equal to negative sine of x.

that the derivative with respect to x of sine of x

That's actually a fairly involved proof that proves this,

but if we assume this, I'm gonna make a visual argument

that this right over here is the derivative

with respect to x of cosine of x is negative sine of x.

So right over here we seen sine of x in red

is showing the derivative, the slope of the tangent line

And we've got an intuition for that in previous videos.

and I'm also gonna shift the blue graph to the left

Well the blue graph is gonna look like this one

right over here and if it was cosine of x up here,

y is equal to cosine of x plus pi over two.

cosine of x, shifted to the left by pi over two.

And this is y is equal to sine of x plus pi over two.

So it should still be the case that the derivative

that the derivative with respect to x of the red graph,

That that is equal to cosine of x plus pi over two.

Well that's the same thing as cosine of x.

You can see this red graph is the same thing as cosine of x.

and you can also see in intuitively or graphically

Now what is cosine of x plus pi over two?

Well once again, from our trig identities,

we know that that is the exact same thing

So there you have it, the visual argument.

shift both of these graphs to the left by pi over two,

is equal to cosine of x plus pi over two.

And this is the same thing as saying what we have

and we have a very strong visual argument for this