is get an intuitive sense for what the derivative

with respect to x of sine of x is

and what the derivative with respect to x of cosine of x is.

And I've graphed y is equal to cosine of x in blue

and y is equal to sine of x in red.

We're not going to prove what the derivatives are,

but we're gonna know what they are, get an intuitive sense

and in future videos we'll actually do a proof.

So let's start with sine of x.

So the derivative can be viewed

as the slope of the tangent line.

So for example at this point right over here,

it looks like the slope of our tangent line should be zero.

So our derivative function should be zero at that x value.

Similarly, over here, it looks like the derivative is zero.

Slope of the tangent line would be zero.

So whatever our derivative function is at that x value,

it should be equal to zero.

If we look right over here on sine of x,

it looks like the slope of the tangent line

would be pretty close to one.

If that is the case, then in our derivative function

when x is equal to zero

that derivative function should be equal to one.

Similarly, over here, it looks like

the slope of the tangent line is negative one,

which tells us that the derivative function

should be hitting the value of negative one at that x value.

So you're probably seeing something interesting emerge.

Everywhere, while we're trying to plot

the slope of the tangent line, it seems to coincide

with y is equal to cosine of x.

And it is indeed the case that the derivative of sine of x

is equal to cosine of x.

And you can see that it makes sense,

not just at the points we tried, but even in the trends.

If you look at sine of x here, the slope is one,

but then it becomes less and less and less positive

all the way until it becomes zero.

Cosine of x, the value of the function is one

and it becomes less and less positive

all the way until it equals zero.

And you could keep doing that type of analysis

to feel good about it.

In another video we're going to prove this more rigorously.

So now let's think about cosine of x.

So cosine of x, right over here,

the slope of the tangent line

looks like it is zero.

And so it's derivative function

needs to be zero at that point.

So, hey, maybe it's sine of x.

Let's keep trying this.

So over here, cosine of x,

it looks like the slope of the tangent line is negative one

and so we would want the derivative to go through

that point right over there.

All right this is starting to seem,

it doesn't seem like the derivative of cosine of x

could be sine of x.

In fact, this is the opposite of what sine of x is doing.

Sine of x is at one, not negative one at that point.

But that's an interesting theory,

maybe the derivative of cosine of x is negative sine of x.

So let's plot that.

So this does seem to coincide.

The derivative of cosine of x here looks like negative one,

the slope of a tangent line

and negative sign of this x value is negative one.

Over here the derivative of cosine of x

looks like it is zero

and negative sine of x is indeed zero.

So it actually turns out that it is the case,

that the derivative of cosine of x is negative sine of x.

So these are really good to know.

These are kind of fundamental

trigonometric derivatives to know.

We'll be able to derive other things for them.

And hopefully this video gives you a good intuitive sense

of why this is true.

And in future videos, we will prove it rigorously.