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# Justifying the power rule

Video transcript

What I want to do
in this video is to see whether the
power rule is giving us results that at least
seem reasonable. This is by no means a
proof of the power rule, but at least we'll feel a little
bit more comfortable using it. So let's say that f
of x is equal to x. The power rule tells
us that f prime of x is going to
be equal to what? Well, x is the same thing
as x to the first power. So n is implicitly
1 right over here. So we bring the 1 out front. It'll be 1 times x to
the 1 minus 1 power. So it's going to be 1 times
x to the 0 power. x to the 0 is just 1. So it's just going
to be equal to 1. Now, does that makes
conceptual sense if we actually try to
visualize these functions? So let me actually try
to graph these functions. So that's my y-axis. This is my x-axis. And let me graph y equals x. So y is equal to f of x here. So y is equal to x. So it looks something like that. So y is equal to x. Or this is f of x
is equal to x, or y is equal to this f of
x right over there. Now, actually, let me
just call that f of x just to not confuse you. So this right over
here is f of x is equal to x that I
graphed right over here. y is equal to f of x,
which is equal to x. And now, let me
graph the derivative. Let me graph f prime of x. That's saying it's 1. That's saying it's 1 for all x. Regardless of what x is,
it's going to be equal to 1. Is this consistent
with what we know about derivatives and
slopes and all the rest? Well, let's look
at our function. What is the slope of the tangent
line right at this point? Well, right over here, this
has slope 1 continuously. Or it has a constant slope of 1. Slope is equal to 1
no matter what x is. It's a line. And for a line, the
slope is constant. So over here, the
slope is indeed 1. If you go to this point over
here, the slope is indeed 1. If you go over here,
the slope is indeed 1. So we've got a pretty
valid response there. Now, let's try something
where the slope might change. So let's say I have g of
x is equal to x squared. The power rule tells
us that g prime of x would be equal to what? Well, n is equal to 2. So it's going to be 2
times x to the 2 minus 1. Or it's going to be equal
to 2 x to the first power. It's going to be equal to 2x. So let's see if this
makes a reasonable sense. And I'm going to try to graph
this one a little bit more precisely. Let's see how precisely
I can graph it. So this is the x-axis, y-axis. Let me mark some stuff off here. So this is 1, 2, 3, 4, 5. This is 1, 2, 3, 4. 1, 2, 3, 4. So g of x. When x is 0, it's 0. When x is 1, it is 1. When x is negative 1, it's 1. When x is 2, it is 4. So that puts us right
over there-- 1, 2, 3, 4. Puts us right over there. When x is negative
2, you get to 4. It's a parabola. You've seen this for many years. I put that point a
little bit too high. It looks something like this. Actually, the last two points I
graphed are a little bit weird. So this might be
right over here. So it looks something like this. It looks something like that. And then, when you
come over here, it looks something like that. It's symmetric. So I'm trying my best
to draw it reasonably. So there you go. That's the graph of g of x.
g of x is equal to x squared. Now, let's graph g prime of
x or what the power rule is telling us a g prime of x is. So g prime of x is equal to 2x. So that's just a line that goes
through the origin of slope 2. So it looks something like that. When x is equal to
1, y is equal to 2. When x is equal to 2, y
or g of x is equal to 4. So it looks something like this. Let me try my best to
draw a straight line. It looks something like this. Now, does this make sense? Well, if you just
eyeball it really fast, if you look at this
point right over here-- and you want to think about
the slope of the tangent line. Let me do this in a color that
pops out a little bit more. So the tangent line would
look something like this. So it looks like it has a
reasonably high negative slope. Yeah. It's definitely
a negative slope, and it's a pretty
steep negative slope. For x is equal to negative
2, g prime of negative 2 is equal to 2 times negative 2,
which is equal to negative 4. So this is claiming that
the slope at this point-- so this right over
here is negative 4-- is saying that the
slope of this point is negative 4. m is
equal to negative 4. That looks about right. It's a fairly steep
negative slope. Now, what happens if
you go right over here when x is equal to 0? Well, our derivative--
if you say g prime of 0-- is telling us that the slope of
our original function, g, at x is equal to 0 is 2 times 0 is 0. Well, does that make sense? Well, if we go to our
original parabola, it does indeed make sense. That's the slope of
the tangent line. The tangent line looks
something like this. We're at a minimum point. We're at the vertex. The slope really
does look to be 0. And what if you
go right over here to x equals 2, the slope
of the tangent line? Well, over here, the tangent
line looks something like this. It looks like a fairly
steep positive slope. What is our derivative telling
us based on the power rule? So this is essentially
saying, hey, tell me what the slope
of the tangent line for g is when x is equal to 2. Well, we figured it out. It's going to be 2 times x. It's going to be 2 times
2, which is equal to 4. It's telling us that the
slope over here is 4. And I'm just using m.
m is often the letter used to denote slope. They're saying that the
slope of the tangent line there is 4, which seems
completely, completely reasonable. So I encourage you to
try this out yourself. I encourage you to try
to estimate the slopes by calculating, by
taking points closer and closer around those points. And you'll see that
the power rule really does give you results
that actually make sense.