# Power rule

### From Conservapedia

The power rule allows one to calculate the derivative of a power of a function in terms of the derivative of the function itself. The power rule states that

for all integers *n*. This rule is useful when combined with the chain rule. As an example we can compute the derivative of *f*(*x*) = (sin(*x*))^{n} as

### Proof

The power rule is simple and elegant to prove with the definition of a derivative:

Substituting *f*(*x*) = *x*^{n} gives

The two polynomials in the numerator can be factored out. It is left as a series, since n can be any integer.

Then if the first factor is eliminated:

Now, the "h" can be eliminated. This is important, since the denominator cannot go to zero':

With no "h" in the denominator, the limit can be evaluated by letting h=0.

- :

We see that there are *n* terms, so:

*f*'(*x*) =*n**x*^{n − 1}.

QED