Power rule (with rewriting the expression) | AP Calculus AB | Khan Academy

What we're going to do in this video is get some practice taking derivatives with the power rule.

So let's say we need to take the derivative with respect to x of 1 over x. What is that going to be equal to? Pause this video and try to figure it out.

So at first you might say, how does the power rule apply here? The power rule, just to remind ourselves, it tells us that if we're taking the derivative of x to the n with respect to x, so if we're taking the derivative of that, that's going to be equal to we take the exponent, bring it out front, and we've proven it in other videos. But this does not look like that.

The key is to appreciate that 1 over x is the same thing as x to the negative 1. So this is going to be the derivative with respect to x of x to the negative 1. Now this looks a lot more like what you might be used to. Where this is going to be equal to, you take our exponent, bring it out front, so that's negative 1 times x to the negative 1 minus 1, negative 1 minus 1.

So this is going to be equal to negative x to the negative 2, and we're done.

Let's do another example. Let's say that we're told that f of x is equal to the cube root of x, and we want to figure out what f prime of x is equal to. Pause the video and see if you can figure it out again.

Well, once again, you might say, hey, how do I take the derivative of something like this? Especially if my goal, or if I'm thinking that maybe the power rule might be useful. The idea is to rewrite this as an exponent. If you could rewrite the cube root as x to the one-third power, the derivative you take the one-third, bring it out front, so it's one-third times x to the one-third minus one power.

So this is going to be one-third times x to the one-third minus one is negative two-thirds, negative two-thirds power, and we are done.

Hopefully, through these examples, you're seeing that the power rule is incredibly powerful. You can tackle a far broader range of derivatives than you might have initially thought.

Let's do another example, and I'll make this one really nice and hairy. Let's say we want to figure out the derivative with respect to x of the cube root of x squared. What is this going to be?

Actually, let's just not figure out what the derivative is; let's figure out the derivative at x equals 8. Pause this video again and see if you can figure that out.

Well, what we're going to do is first just figure out what this is, and then we're going to evaluate it at x equals 8. The key thing to appreciate is this is the same thing, and we're just going to do what we did up here as the derivative with respect to x.

Instead of saying the cube root of x squared, we could say this is x squared to the one-third power, which is the same thing as the derivative with respect to x of well x squared. If I raise something to an exponent and then raise that to an exponent, I could just take the product of the exponents.

So this is going to be x to the 2 times 1/3 power or to the two-thirds power. Now this is just going to be equal to, I'll do it right over here, bring the two-thirds out front: two-thirds times x to the... What's two-thirds minus one? Well, that's 2/3 minus 3/3, or it would be negative 1/3 power.

We want to know what happens at x equals 8. So let's just evaluate that. That's going to be two-thirds times x, which is equal to 8 to the negative 1/3 power. Well, what's 8 to the 1/3 power? 8 to the 1/3 power is going to be equal to 2.

So 8 to the negative 1/3 power is one-half. Actually, let me just do that step by step. So this is going to be equal to two-thirds times we could do it this way: one over eight to the one-third power, and so this is just one over two.

Two-thirds times one-half, well, that's just going to be equal to one-third, and we're done.