- [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.
And we're gonna base this argument,
based on a previous proof we made
that the derivative with respect to x of sine of x
is equal to cosine of x.
So we're gonna assume this over here.
I encourage you to watch that video.
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
and we see cosine of x in blue.
And we're assuming that this blue graph
is showing the derivative, the slope of the tangent line
for any x value of the red graph.
And we've got an intuition for that in previous videos.
Now what I'm gonna do next,
is I'm gonna shift both of these graphs
to the left by pi over two.
Shift it to the left by pi over two
and I'm also gonna shift the blue graph to the left
by pi over two.
And so what am I going to get?
Well the blue graph is gonna look like this one
right over here and if it was cosine of x up here,
we can now say that this is equal to
y is equal to cosine of x plus pi over two.
This is the blue graph,
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.
Now the visual argument is, all I did,
is I shifted both of these graphs
to the left by pi over two.
So it should still be the case that the derivative
of the red graph is the blue graph.
So we should still be able to say
that the derivative with respect to x of the red graph,
sin of x plus pi over two
that that is equal to the blue graph.
That that is equal to cosine of x plus pi over two.
Now what is sin 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.
We know that from our trig identities
and you can also see in intuitively or graphically
just by looking at these graphs.
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
as negative sine of x.
So there you have it, the visual argument.
Just start with this knowledge,
shift both of these graphs to the left by pi over two,
it should still be true,
that the derivative with respect to x
of sine of x plus pi over two
is equal to cosine of x plus pi over two.
And this is the same thing as saying what we have
right over here.
So now we should feel pretty good.
We proved this in a previous video
and we have a very strong visual argument for this
for cosine of x in this video.