Name: Product rule proof | Taking derivatives | Differential Calculus | Khan Academy
Uploaded: 2022-07-12T12:49:59.000Z
Duration: 9 min 25 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...

To-infinitive

- [Voiceover] What I hope to do in this video

is give you a satisfying proof of the product rule.

Present continuous

So let's just start with our definition of a derivative.

and if I wanted to take the derivative of it,

this is the slope of the tangent line and all of that,

And if I can come up with a simple thing for this,

Well if we just apply the definition of a derivative,

and the denominator, I'm gonna write a big,

And then I'm gonna evaluate this thing at X plus H.

G of X plus H and from that I'm gonna subtract

And I'm gonna put a big, awkward space here

All I did so far is I just applied the definition

of the derivative, instead of applying it to F of X,

Now why did I put this big, awkward space here?

Because just the way I've written it write now,

it doesn't seem easy to algebraically manipulate.

there doesn't seem to be anything obvious to do.

I guess you could view it as a little bit of a trick.

I can't claim that I would have figured it out on my own.

Maybe eventually if I were spending hours on it.

I'm assuming somebody was fumbling with it long enough

"Look, if I just add and subtract at the same term here,

"I can begin to algebraically manipulate it

I've got to add it too, so I don't change the value

I just added and subtracted the same thing,

in interesting algebraic ways to get us to

I'm gonna look at, I'm gonna look at this part,

And in particular, let's see, I am going to

you're going to be left with G of X plus H.

G of, that's a slightly different shade of green,

I got a new software program and it's making it hard

Subtitles ListPlay Video

Product rule proof | Taking derivatives | Differential Calculus | Khan Academy

awkward

essentially

assume

eventually