Name: Basic derivative rules: table | Derivative rules | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T12:31:21.000Z
Duration: 9 min
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

For f, they tell us for given values of x

in terms of this kind of absolute value expression.

What we're curious about is what is the derivative

with respect to x, of h of x at x is equal to nine.

Another way just to get familiar with the notation

of writing this, the derivative of h of x

This is equivalent to h, we need that blue color,

and the prime signifies that we're taking the derivative.

so h prime of nine is what this really is.

Actually I'm going to do this in a different color.

to figure out what the derivative with respect to x of h is.

We get a derivative, I'll do that same white color.

is going to be equal to the derivative with respect to x

I could actually just, well I'll just rewrite it.

Three times f of x, plus two times g of x.

as the sum of the derivatives of each of the terms.

of three times, I'll write that a little bit neater.

Three times f of x, plus the derivative with respect to x

or I guess you could say a scaling factor times a function.

The derivative of a scalar times the function

Well that just means that this first term right over here

three times the derivative with respect to x

plus this part over here is the same thing as two.

Okay, make sure I don't run out of space here,

plus two times the derivative with respect to x.

The derivative with respect to x of g of x.

plus two times the derivative of g with respect to x.

If we want to write it in this kind of prime notation here,

we could rewrite it as h prime of x is equal to

It's three times f prime of x, plus two times g prime of x.

Once you are more fluent with this property,

The derivative of a scalar times something

is the same thing as a scalar times the derivative

You really could have gone straight from here

Well they tell us, when x is equal to nine,

but more importantly f prime of nine is three.

This part right over here evaluates that part's three.

Let's look at this function a little bit more closely.

There's a couple of ways we could think about it.

and do this right over here is our x-axis.

Now when does an absolute value function like this,

when is this going to hit a minimum point?