Name: Power rule (with rewriting the expression) | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T12:22:56.000Z
Duration: 4 min 23 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...

Future simple

- [Instructor] What we're going to do in this video

is get some practice taking derivatives with the power rule.

The imperative

Pause this video and try to figure it out.

The power rule, just to remind ourselves,

the derivative of x to the n with respect to x,

so if we're taking the derivative of that,

we take the exponent, bring it out front,

and the key is to appreciate that one over x

is the same thing as x to the negative one.

with respect to x of x to the negative one.

you take our exponent, bring it out front,

and we wanna figure out what f prime of x is equal to.

Pause the video and see if you can figure it out again.

"Hey, how do I take the derivative of something like this,

"especially if my goal or if I'm thinking that maybe

And the idea is to rewrite this as an exponent,

if you can rewrite the cube root as x to the 1/3 power.

And so, the derivative, you take the 1/3,

you're seeing that the power rule is incredibly powerful.

You can tackle a far broader range of derivatives

and I'll make this one really nice and hairy.

Let's say we wanna figure out the derivative

let's just not figure out what the derivative is,

let's figure out the derivative at x equals eight.

Pause this video again and see if you can figure that out.

and then we're going evaluate it at x equals eight.

And the key thing to appreciate is this is the same thing,

and we're just gonna do what we did up here

Instead of saying the cube root of x squared,

we can say this is x squared to the 1/3 power,

which is the same thing as the derivative

well, x squared, if I raise something to an exponent

I can just take the product of the exponents.

And so, this is gonna be x to the two times 1/3 power

And now, this is just going to be equal to,

I'll do it right over here, bring the 2/3 out front,

2/3 times x to the, what's 2/3 minus one?

Now, we wanna know what happens at x equals eight,

That's going to be 2/3 times x is equal to eight

Eight to the 1/3 power is going to be equal to two,

and so, eight to the negative 1/3 power is 1/2.

Actually, let me just do that step-by-step.

So, this is going to be equal to 2/3 times,

we could do it this way, one over eight to the 1/3 power.

well, that's just going to be equal to 1/3, and we're done.