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## AP®︎/College Calculus AB

# Power rule (with rewriting the expression)

AP.CALC:

FUN‑3 (EU)

, FUN‑3.A (LO)

, FUN‑3.A.1 (EK)

We can use the power rule to find the derivatives of functions like 1/x, ∛x, or ∛x². To do that, we first need to rewrite those functions as xⁿ, where n would be negative or a fraction.

## Video transcript

- [Instructor] What we're
going to do in this video is get some practice taking
derivatives with the power rule. So let's say we take the derivative with respect to x of one 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 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 is gonna be n times x to the, and then we decrement the exponent. So, n minus one. But this does not look like that, and the key is to
appreciate that one over x is the same thing as
x to the negative one. So, this is going to be the derivative with respect to x of
x to the negative one. And 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 it's negative one, times x to the negative one minus one, negative one minus one. And so, this is going to be equal to negative x to the negative two, 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 wanna 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?" 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, bring it out front, so it's 1/3 x to the 1/3 minus one power. And so, this is going to be 1/3 times x to the 1/3 minus one is negative 2/3, negative 2/3 power, and we are done. And 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 wanna
figure out the derivative with respect to x of the cube root of x squared. What is this going to be? And actually, 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. Well, what we're gonna do is first just figure out what this is 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 as the derivative with respect to x. 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 with respect to x of, well, x squared, if I raise
something to an exponent and then raise that 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 or to the 2/3 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? Well, that's 2/3 minus 3/3 or it would be negative 1/3 power. Now, we wanna know what
happens at x equals eight, so let's just evaluate that. That's going to be 2/3
times x is equal to eight to the negative 1/3 power. Well, what's eight to the 1/3 power? 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. And so, this is just one over two, 2/3 times 1/2, well, that's just going to be
equal to 1/3, and we're done.