Name: 08-1 sin(x) 和 cos(x) 的導數（Derivatives of sin(x) and cos(x) | Derivative rules | AP Calculus AB | Khan Academy）
Uploaded: 2022-07-12T12:31:28.000Z
Duration: 3 min 41 s
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- [Instructor] What I'd like to do in this video

is get an intuitive sense for what the derivative

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

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 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.

So whatever our derivative function is at that x value,

it looks like the slope of the tangent line

If that is the case, then in our derivative function

that derivative function should be equal to one.

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.

the slope of the tangent line, it seems to coincide

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

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

Cosine of x, the value of the function is one

And you could keep doing that type of analysis

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

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

and so we would want the derivative to go through

it doesn't seem like the derivative of cosine 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.

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

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

and negative sign of this x value is negative one.

So it actually turns out that it is the case,

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

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

And hopefully this video gives you a good intuitive sense

And in future videos, we will prove it rigorously.