Using mathematical induction prove that `d/(dx)(x^n)=n x^(n-1)` for all positive integers n.

Explanation

Let `P(n) : (d)/(dx)(x^(n))= n*x^(n-1)`
For `n=1`
`P(1) :(d)/(dx)(x)= 1*x^(1-1)=1`
which is true.
` :. `P(n) is true for `n=1`
Let P(n) be true for `n=k.`
` :. P(k) :(d)/(dx)(x^(k))= k*x^(k-1) " " `...(1)
For `n=k+1`
` P(k+1) :(d)/(dx)(x^(k+1))= (d)/(dx)(x^(k)*x)`
`=x*(d)/(dx)(x^(k))+x^(k)*(d)/(dx)(x)`
`=x*k*x^(k-1)+x^(k)*1`
`=kx^(k)+x^(k)=(k+1)x^(k)` From equation (1)
`implies P(n)` is also true for `n=k+1.`
Hence, from the principle of mathematical induction, P(n) is true for all natural numbers n.