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AP Calc: FUN‑3 (EU), FUN‑3.A (LO), FUN‑3.A.1 (EK)

In this video, we will
cover the power rule, which really simplifies
our life when it comes to taking
derivatives, especially derivatives of polynomials. You are probably
already familiar with the definition
of a derivative, limit is delta x
approaches 0 of f of x plus delta x minus f of
x, all of that over delta x. And it really just
comes out of trying to find the slope of a tangent
line at any given point. But we're going to see
what the power rule is. It simplifies our life. We won't have to take these
sometimes complicated limits. And we're not going to
prove it in this video, but we'll hopefully get
a sense of how to use it. And in future videos, we'll get
a sense of why it makes sense and even prove it. So the power rule just tells us
that if I have some function, f of x, and it's equal
to some power of x, so x to the n power, where
n does not equal 0. So n can be anything. It can be positive, a
negative, it could be-- it does not have to be an integer. The power rule tells us that
the derivative of this, f prime of x, is just going
to be equal to n, so you're literally bringing
this out front, n times x, and then you just decrement
the power, times x to the n minus 1 power. So let's do a couple
of examples just to make sure that that
actually makes sense. So let's ask ourselves,
well let's say that f of x was equal to x squared. Based on the power
rule, what is f prime of x going to be equal to? Well, in this
situation, our n is 2. So we bring the 2 out front. 2 times x to the
2 minus 1 power. So that's going to be 2 times
x to the first power, which is just equal to 2x. That was pretty straightforward. Let's think about
the situation where, let's say we have g of x is
equal to x to the third power. What is g prime of x going
to be in this scenario? Well, n is 3, so we just
literally pattern match here. This is-- you're
probably finding this shockingly straightforward. So this is going to be 3 times
x to the 3 minus 1 power, or this is going to be
equal to 3x squared. And we're done. In the next video
we'll think about whether this
actually makes sense. Let's do one more
example, just to show it doesn't have to
necessarily apply to only these kind
of positive integers. We could have a
scenario where maybe we have h of x. h of x is equal
to x to the negative 100 power. The power rule tells
us that h prime of x would be equal to what? Well n is negative 100,
so it's negative 100x to the negative
100 minus 1, which is equal to negative
100x to the negative 101. Let's do one more. Let's say we had z of x. z of x is equal to x
to the 2.571 power. And we are concerned with
what is z prime of x? Well once again, power
rule simplifies our life, n it's 2.571, so
it's going to be 2.571 times x to the
2.571 minus 1 power. So it's going to
be equal to-- let me make sure I'm not falling
off the bottom of the page-- 2.571 times x to
the 1.571 power. Hopefully, you enjoyed that. And in the next few
videos, we will not only expose you to more
properties of derivatives, we'll get a sense for why
the power rule at least makes intuitive sense. And then also prove the
power rule for a few cases.

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