Power rule: d(xⁿ)/dx = nx^n-1

There are really three proofs which we can provide here and we are going to suffer all three here therefore you can notice all of them.

The proof of this theorem will work for any real number n. Though, this does suppose that you've read most of the previous section and so must only be read after you have gone during the whole section.

Proof

In this proof we no longer require to confine n to be a positive integer. This can here be any real number. Though, this proof also supposes which you have read all the way throughout previous section. In particular this requires both Implicit Differentiation and Logarithmic Differentiation. If you have not read, and know, these sections so it proof will not make any sense to your understanding.

Therefore, to find set up for logarithmic differentiation let's first name

 y = xⁿ

 after that take the log of both sides and simplify the right side by using logarithm properties and after that differentiate by using implicit differentiation.

 ln y = ln xn

ln y = n ln x

y′/y = n (1/x)

At last, all we require to do is solving for y′ and after that substitute into for y.

y' = y(n/x) = xⁿ(n/x) = nx^n-1

Before going onto the subsequent proof, let's consider that in all three proofs we did need the exponent n, be a number which is integer, any real number in this proof.

At last, in the third proof we would have found various derivatives if n had not been a constant.

It is significant as people will frequently misuse the power rule and utilize this even while the exponent is not a number or/and the base is not a variable.