Chapter A2, Problem 3E

To determine

To show: $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ for every positive integer $n$ .

Explanation of Solution

Proof:

The principle of mathematical induction states that if a certain mathematical relation is true for a base integer and, thereafter, assuming the relation to be true for an integer $k$ greater than the base integer, if we can prove the relation to be true for $k+1$ , the relation is true for all integers greater than the base integer.

Step 1: The formula holds for $n=1$ because $\frac{d\left(x^1\right)}{dx}=1$

Step 2: Suppose $\frac{d\left(x^k\right)}{dx}=k{x}^{k-1}$

Then

  $\begin{array}{l}\frac{d\left(x^{k+1}\right)}{dx}=\frac{d\left(x\cdot x^k\right)}{dx}\\ =x\frac{d\left(x^k\right)}{dx}+x^k\frac{d\left(x\right)}{dx}\\ =x\cdot k{x}^{k-1}+x\\ =k{x}^k+x\\ =\left(k+1\right)x^k\end{array}$

Hence, by principal of mathematical induction, this relation is true for all $n$ .

Conclusion:

By principal of mathematical induction, this relation is true for all $n$ .