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### Course: AP®︎/College Calculus BC > Unit 2

Lesson 12: Optional videos- Proof: Differentiability implies continuity
- Justifying the power rule
- Proof of power rule for positive integer powers
- Proof of power rule for square root function
- Limit of sin(x)/x as x approaches 0
- Limit of (1-cos(x))/x as x approaches 0
- Proof of the derivative of sin(x)
- Proof of the derivative of cos(x)
- Product rule proof

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# Justifying the power rule

Let's explore the power rule's validity by examining the derivatives of x¹ and x². We'll analyze the slopes of tangent lines for these functions and then see how the power rule provides reasonable results, building our confidence in its usefulness. Created by Sal Khan.

## Want to join the conversation?

- when we say slope =4 what do we actually mean? Because when I think of slope I think of degrees(17 votes)
- When I think of slope, I think of rise/run. You may remember hearing your teacher say "rise over run". For slope = 4, I think to myself slope = 4 = rise/run = 4/1. That tells me for every 4 units up, I go over 1.(6 votes)

- How about if there was a square root of x as the denominator of a fraction? For example: f(x) = 1/√x *Please help, and thank you : ) *(7 votes)
- When taking the derivative of √x, it's a good idea to rewrite it as √x = x^(1/2). So when you have 1/√x, you would write it as x^(-1/2). Now you can just use the power rule:

Derivative of 1/√x = Derivative of x^(-1/2)

=(-1/2)x^(-1/2 -1) = (-1/2)x^(-3/2)(26 votes)

- Given f(x) = x, is f'(0) undefined, as x^0 is defined only if x != 0?(7 votes)
- Ok, I figured out that in this case we need to fall back on the derivative definition:
`f'(x) = lim h->0 (f(x+h)-f(x))/h = lim h->0 (x+h-x)/h = lim h->0 h/h = lim h->0 1 = 1`

This way f'(x) is defined even for x=0(16 votes)

- Does the power rule work with polynomials like 2x^3+4x^2+9x+17?(4 votes)
- There is a theorem that states that the derivative of a sum is equal to the sum of the derivatives. So yes! Just use the power rule on each term separately, and then add them all up.(21 votes)

- I know it late to ask but I just wanted to ask that is f(x) always equal to the y? Even in 3-D?(12 votes)
- In the two dimensional case we typically label the dependent variable Y, meaning it's value is dependent on other variables, in 2D there is only one other variable, usually called X (an "independent variable". There is nothing special about their labels howevere, and you could swap X and Y or even solve for Y for instance x = (y - b)/m for slope intercept and graph that way. The X and Y of the graph are just traditional standards, it is more important to understand how dependent and independent variables relate to eachother.

In three dimensions, if solving for Y, X and Z are would normally both be dependent variables(6 votes)

- So if n is 0 and f(x) = x, would f'(x) = 1/x?(3 votes)
- No. The derivative of x is 1. If f(x) was x^0 then the power rule states that you would pull the 0 out in front and subtract one from the exponent, well, anything multiplied by 0 is still 0.(11 votes)

- This may be a tad off topic... I apologize in advance.

If you take any function f(x) and take the derivative of the derivative of the derivative... etc. Where will you end up? If you keep going to infinity, will the derivative just be a horizontal line?(5 votes)- Hi Natalie,

Depends on the function. Some of them are interesting for example, Google the derivative of f(x) = e^x

For the types of function that you were likely thinking about I have a few questions that will lead you to the answer.

1) What is the derivative of a constant?

2) Is zero a constant?

Regards,

APD(4 votes)

- After we make the curve on the graph , what does making a seacant line represents ?

I mean like in the above video what is the use of makin a seacant line after makig the curve . The slope of any point can also be calculated without makin the seacant line .(3 votes)- From what I have learned we would use the Secant line for the average slope of a curve between two points and then use the Tan Line as h->0 to get the instantaneous rate of change of the curve at any given point as h->0 Look at the video for Diveratives and it shows a lot more information about the difference of the secant line and Tan Line.

Also one key note the steps to get the secant line and tan line are very similer the only difference is you take the LIM as h-> 0 to get the tan line.(3 votes)

- How would the power rule apply to a question that asks for you to simplify and answer using a single exponent of something like this

(v^6)^5(3 votes)- You multiply the exponents.

(v⁶)⁵ = v³⁰

Then you apply the power rule in the normal way.

Thus,

d/dx (x⁶)⁵ = 30x²⁹

So a lot of these problems are easier if you simplify before you differentiate. It doesn't always help, but it often does.(3 votes)

- Can anyone proof (origin) of this rule ??(2 votes)
- Are you familiar with the Binomial theorem? If not, I recommend checking it out first.

https://www.khanacademy.org/math/differential-calculus/taking-derivatives/power_rule_tutorial/v/proof-d-dx-x-n

https://www.khanacademy.org/math/differential-calculus/taking-derivatives/power_rule_tutorial/v/proof-d-dx-sqrt-x(3 votes)

## 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.