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
The power rule is simple and elegant to prove with the definition of a derivative:
Substituting f(x) = xn 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) = nxn − 1.