For all integers $n\geq 1$ the derivative of the power function is ${{d}\over{dx}}x^{n}=nx^{n-1}.$ (Later we shall prove that this formula works for all real powers.)

Proof by induction. Let $P_{n}$ be the statement ${{d}\over{dx}}x^{n}=nx^{n-1}.$

Basis of induction.$P_{1}$ asserts that ${{dx}\over{dx}}=1$, which is true.

Induction step. Suppose that $P_{n}$ holds for some $n$, and consider $P_{n+1}$. By the product rule, we have

hence result by induction.