If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Proof of power rule for positive integer powers

We dive into proving the formula for the derivative of x^n by skillfully applying the binomial theorem. Together, we expand (x + Δx)^n, simplify the expression, and take the limit as Δx approaches zero to reveal the power rule for derivatives. Created by Sal Khan.

## Want to join the conversation?

• In the second step, I thought the definition of a derivative was (f(x+∆x)-f(x))/∆x. But in this video it shows ((x+∆x)^n-x^n)/∆x. Why is this?
• Using f(x) notation generalized the definition of the derivative. Now, we are plugging in what are actual f(x) is. In this case, f(x)=x^n so f(x+∆x)=(x+∆x)^n and so on.
• What does the limit (delta)x approaches zero mean? Why does Sal say that (delta)x approaches zero?
• Delta X is the change in X, to take the average slope of a function (over the range of delta x) you divide delta Y by Delta X.

To take the slope of a function at any point (i.e. the derivative) you take the limit as delta X approaches 0.
• Why is the derivative of x^n = nx^n-1, I mean I know Sal proved it above but I was wondering since the answer is so neat is there a physical or intuitive reason as to why it is so neat?
• Because you are looking at a tanget line of a very well defined curve. That makes it nice and neat.
• When Sal is cancelling out all the terms while he is taking the limit, I noticed that if n=1 then that last term would be ∆x^0. But since he was turning all the ∆x's into zero, if n=1 then the last term would be 0^0, which is undefined, so...how does that work out?
• *That's a good question.*
Well, ∆x is tending to zero, which means it is infinitely small, but not 'zero'. It might seem confusing, but it's sort of like predicting what would happen if ∆x became zero (by taking ∆x to be something that's really really really ... (infinite times) ... really small).
• Where can I find Leibniz Rule in KHANACADEMY.ORG's library?
• How would you differentiate x^x w.r.t. x?
In case of x^x, both the base and the index are variables
So do we use dx^n/dx or do we use d(a^x)/dx?
• Neither. Write it as exp(ln(x^x)) = exp(x ln x).
(exp(x) is e^x, if you haven't seen it before.)
• For those of us who skipped the Binomial Theorem in the Algebra II playlist because they don't know any combinatorics, how important is this proof? Can it be safely skipped for the time being? Building up one's knowledge of combinatorics to the point where one can understand and apply the Binomial Theorem feels like a lot of work...
• You don't have to know any combinatorics to understand the binomial formula - it is purely algebraic, it just happens to have a combinatorial interpretation as well. I am sure you can find a proof by induction if you look it up.

What's more, one can prove this rule of differentiation without resorting to the binomial theorem. For instance, using induction and the product rule will do the trick:

Base case n = 1
d/dx x¹
= lim (h → 0) [(x + h) - x]/h
= lim (h → 0) h/h
= 1.

Hence d/dx x¹ = 1x⁰.

Inductive step
Suppose the formula d/dx xⁿ = nxⁿ⁻¹ holds for some n ≥ 1. We will prove that it holds for n + 1 as well. We have xⁿ⁺¹ = xⁿ · x. By the product rule, we get

d/dx xⁿ⁺¹
= d/dx (xⁿ · x)
= [d/dx xⁿ]·x + xⁿ·[d/dx x]
= nxⁿ⁻¹ · x + xⁿ · 1
= nxⁿ + xⁿ
= (n + 1)xⁿ.

This completes the proof.

There is yet another proof relying on the identity

(bⁿ - aⁿ)
= (b - a)[bⁿ⁻¹ + bⁿ⁻²a + bⁿ⁻³a² + … + b²aⁿ⁻³ + baⁿ⁻² + aⁿ⁻¹].

(To prove this identity, simply expand the right hand side, and note that most of the terms will cancel - alternatively, prove it by induction.)

Letting b = x+h and a = x in the formula above, one gets

(x+h)ⁿ - xⁿ
= h[(x+h)ⁿ⁻¹ + x(x+h)ⁿ⁻² + … + xⁿ⁻²(x+h) + xⁿ⁻¹].

Now divide by h and apply some induction to arrive at the desired conclusion.
• isn't it simpler to use power rule
• This is supposed to be a proof of the power rule. Using the power rule to prove the power rule would be circular reasoning. Also, one can not logically use the power rule without first proving it.