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In this video, we will cover the power rule,

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which really simplifies our life,
when it comes to taking derivatives,

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especially derivatives of polynomials.

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You're probably already familiar
with the definition of derivatives.

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The limit as delta x approaches 0,

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f(x+delta x) - f(x),
all of that over delta x.

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and it really just comes out of
trying to find the slope of a tangent line,

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at a any given point.

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What we're going to see with the
power rule is it simplifies our life.

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We won't have to take these,
sometimes complicated, limits.

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We're not going to prove it
in this video,

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we'll hopefully get a
sense of how to use it,

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and in future videos,
we'll get a sense of why it makes sense,

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and even prove it.

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So the power rule just tells us,

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if I have some function,
f(x), equal to some power of x,

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so, x^n, where n is not 0.

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so n could be anything,
it could be positive and negative,

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it does not have to be an integer.

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The power rule tells us that
the derivative of this,

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f'(x), is just equal to n,
so you're literally bringing this out front,

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n * x, and then you just
decrement the power.

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so f'(x) = n*x^(n-1).

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So let's do a couple of examples,
just to make sure that makes sense.

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So let's ask ourselves,
let's say f(x)=x^2.

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Based on the power rule,
what is f'(x) going to be equal to?

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Well, in this situation,
our n=2,

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so we bring the 2 out front,
2*x^(2-1).

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That will be 2*x^1,
which is just equal to 2x.

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That was pretty straightforward.

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Let's think about the situation
where g(x)=x^3.

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What is g'(x) going to be,
in this scenario?

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Well, n is 3, so we just
literally pattern match here,

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you'll find this to be
shockingly straightforward,

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so this is going to be
3*x^(3-1) = 3x^2.

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And we're done!

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In the next video, we will think about
whether this actually makes sense.

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Let's do one more example,

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just to show it doesn't have to
necessarily apply to only the positive integers.

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We could have a scenario where
we could have h(x)=x^(-100).

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The power rule tells us that
h'(x) would be equal to what?

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Well n = -100, so it's
-100 x^(-100-1).

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which is equal to -100x^(-101).

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Let's do one more.

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Let's say we had z(x)=x^2.571.

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And we are concerned with
what is z'(x).

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Once again, power rule simplifies our life,
so this is going to be

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2.571 * x^(2.571 - 1),
so it's going to be equal to,

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and make sure I'm not falling off
the bottom of the page,

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2.571*x^1.571.

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Hopefully you enjoyed that.

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In the next video, we will
not only expose you to

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more properties of the derivatives,
we'll get a sense for why

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the power rule at least makes 
intuitive sense,

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and then also prove the power rule,
for few cases.