Using mathematical induction prove that

To prove: $\mathrm{P}(n): \frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$ for all positive integers $n$

For n = 1,

$\mathrm{P}(1): \frac{d}{d x}(x)=1=1 \cdot x^{1-1}$

$\therefore \mathrm{P}(n)$ is true for $n=1$

Let P(k) is true for some positive integer k.

That is, $\mathrm{P}(k): \frac{d}{d x}\left(x^{k}\right)=k x^{k-1}$

It has to be proved that P(k + 1) is also true.

Consider $\frac{d}{d x}\left(x^{k+1}\right)=\frac{d}{d x}\left(x \cdot x^{k}\right)$

$=x^{k} \cdot \frac{d}{d x}(x)+x \cdot \frac{d}{d x}\left(x^{k}\right)$    [By applying product rule]

$=x^{k} \cdot 1+x \cdot k \cdot x^{k-1}$

$=x^{k}+k x^{k}$

$=(k+1) \cdot x^{k}$

$=(k+1) \cdot x^{(k+1)-1}$

Thus, P(k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, the statement P(n) is true for every positive integer n.

Hence, proved.