Name: Product rule example
Uploaded: 2022-07-12T12:40:53.000Z
Duration: 4 min 28 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...

- [Voiceover] So let's see if we can find

We know the derivative with respect to x of

We know how to find the derivative cosine of x.

The derivative with respect to x of cosine of x

But, how do we find the derivative of their product?

Well as you can imagine, this might involve

So if we take the derivative with respect to x

so this is, we could call this u of x times

This is going to be equal to the derivative

times just the second expression not the derivative of it,

the first expression, not its derivative,

U of x times the derivative of the second expression.

Times the derivative of the second expression.

you're going to end up with two different terms.

the derivative of one of them, but not the other one,

and then the other one you'll take the derivative of

So, the derivative of u times v is u prime times v,

that might even be a little bit confusing,

but that's why we have a tangible example here

We could say that u of x is equal to e to the x.

That's one of the most magical things in mathematics.

One of the things that makes e so special.

So u prime of x is still equal to e to the x.

So, the derivative of e to the x which is just,

not taking it's derivative, so times cosine of x.

not taking its derivative, so e to the x,

times the derivative of the second expression.

because e to the x is its own derivative.

This right over here, you can view this as

That's what's exciting about that expression,

And then this is just e to the x without taking

it's derivative - they are of course, the same thing.

We could write either as e to the x times cosine of x,

e to the x times cosine of x minus sine of x.

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Product rule example

expression

negative

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imagine