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

Lesson 5: Applying the power rule# Power rule

Let's dive into the power rule, a handy tool for finding the derivative of xⁿ. This rule simplifies the process of taking derivatives, especially for polynomials, by bringing the exponent out front and decrementing the power. We explore examples with positive, negative, and fractional exponents. Created by Sal Khan.

## Want to join the conversation?

- Does the power rule tell us how to deal with an expression with a coefficient, like f(x) = 2x^3? And what do I do if I have more than one term in my equation, like f(x) = 3x^2 + x + 3?(98 votes)
- For composed function, you would use the chain rule: https://www.khanacademy.org/math/differential-calculus/taking-derivatives/chain_rule/v/chain-rule-introduction(2 votes)

- What would the derivative be of something like 2^X? (two raised to the power of x) Would it still be 2x?(20 votes)
- No.

The Power Rule is for taking the derivatives of polynomials, i.e. (4x^5 + 2x^3 + 3x^2 + 5). All the terms in polynomials are raised to integers.

2^x is an exponential function not a polynomial.

The derivate of 2^x is ln(2)*2^x, which you would solve by applying the Derivative of Exponential Rule: The derivative of an exponential function with a base of C is the natural log of C times the exponential function.

Derivate of C^x = ln(C) * C^x

In this case, C = 2. So... derivate of 2^x = ln(2) * 2^x

Sal does a a proof for common functions, in one of the later tutorials that probably walks you through a rigorous proof of it. (I haven't seen it yet).(58 votes)

- Why can't n = 0? If n is 0, then x^n is 1, right? Then its derivative is 0 like any other constant. And that follows the power rule doesn't it? So why can't n = 0?(24 votes)
- There is no reason why n can't be 0 for the power rule in differentiation. The reason why we say this is because this is more convenient when we reverse the power rule when calculating antiderivatives. For example, if I told you dy/dx=6x^2, with the power rule reversed we can show that y=2x^3. This is not possible with dy/dx=1/x, as we would be dividing by zero.

For differentiation, n can be 0. There is nothing wrong with it.(40 votes)

- At0:58, can n be imaginary?(14 votes)
- Yes, it can. But, of course, working with complex exponents is a bit difficult, although the power rule still applies.

Thus,

d/dx 5x^(3i) = 15𝑖x^(-1+3𝑖)(17 votes)

- Is there ever a case where you take a derivative twice?

So for example, x^3-5x^2+12.

Using the power rule, you'd get 3x^2-10x

Is there a case where you would apply the power rule again and get 6x-10?(9 votes)- Definitely. The 2nd derivative is used frequently. One popular use is to find the nature of stationary points on a curve (that means whether they're minimum or maximum). The interpretation is the rate of change of the slope. Is the slope increasing or decreasing?

You might want to peek at this video: https://www.khanacademy.org/math/calculus/derivative_applications/critical_points_graphing/v/concavity--concave-upwards-and-concave-downwards-intervals. The screenshot of the video by itself is helpful. It shows the main curve at the top, the first derivative underneath and then the second derivative at the bottom. It's interesting to see how they relate.(11 votes)

- Does the power rule work If you have a function with a square root or a fraction?(6 votes)
- Yes. For example, if you have square root of x as f(x) which is x^1/2, you use the power rule to get 1/2*x^-1/2 which is just 1 divided by (2*square root of x). Sorry, I don't know how to notate square roots on KA, if it's even possible.(6 votes)

- The test questions to this lesson include the following example:

a= 2x^4 + 6x^3 - 7x^2

and gives the answer (a prime) = -14x

Why would the answer not also include the terms: 8x^3 + 18x^2?(5 votes)- I'm not sure why because a' should equal 8*x^3+18*x^2-14*x. What does a mean in this problem's case. Is it referring t f(x) or what?(2 votes)

- When asked for a proof of a derivative of a constant, can the power rule be used as where c represents the constant and x the variable:

y=cx^0

y'=c*0*x^-1

Because there the entire term is multiplied by zero, the expression for the derivative is equal to 0?(4 votes)- I think that would be adequate, but you can more directly prove it from the definition of a derivative: since for a constant f(x) is the same value for all x , you have a 0 in the numerator. Thus, the derivative of all constants must be 0.(2 votes)

- How do I figure out if a number can be expressed as a power greater than 1? Example 120, 400, 100, 250, 200 and how do you express the number as a power?(3 votes)
- You could factor each number and look for factors that occurs an even number of times.

100 = 2 * 2 * 5 * 5 = (2 * 5) * (2 * 5) = 10 * 10 = 10^2

400 = 2 * 2 * 2 * 2 * 5 * 5 = (2 * 2 * 5) * (2 * 2 * 5) = 20 * 20 = 20^2

Some numbers will give you several options, e.g. 81 = 3^4 = 9^2.(3 votes)

- I watched in a video example x to the root 5 power. and the answer was root 5 * x ^ (root5-1). But I thought this wasn't allowed since root 5 is irrational and the power rule only applies to rational exponents. Can someone please explain? Thank you!(3 votes)
- As you proceed through the calculus curriculum, the power rule can be proved to apply to a larger and larger subset of exponents. I'm not sure if Sal specifically proves it, but using implicit differentiation you can prove that the power rule works for all real exponents.(2 votes)

## Video transcript

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.