 Hi, and welcome to the session. I am Deepika here. Let's discuss the question using mathematical induction Prove that d by dx of x is to power n is equal to nxn minus 1 for all positive integers n. So let's start the solution. Given statement is, d by dx of x is to power n is equal to nxn minus 1. We have to prove that this statement is true for all positive integers n. And this we have to prove by using mathematical induction. So let us start for n is equal to 1. We have d by dx of x is to power 1 is equal to 1 into x is to power 0, which is equal to 1. Here d by dx of x is to power 1 is equal to 1. Hence statement holds for n is equal to 1. Let us assume that statement is true for n is equal to k. That is, d by dx of x is to power k is equal to k into x is to power k minus 1. So let us give this equation as number 1. Now we will prove that statement is true for n is equal to k plus 1. So let's, for n is equal to k plus 1. Now d by dx of x is to power k plus 1 is equal to d by dx of x is to power k into x. So this is equal to x is to power k into d by dx of x plus x into d by dx of x is to power k. So this we have obtained by the product's rule. This is again equal to x is to power k into 1 plus x into now use equation 1. The derivative of x is to power k is k into x is to power k minus 1. So this is k into x is to power k minus 1 using equation 1. So this is equal to x is to power k plus k into x is to power k minus 1 plus 1. And this is equal to x is to power k plus k into x is to power k, which is equal to x is to power k into 1 plus k. This can be written in the form. 1 plus k into x is to power 1 plus k minus 1. Therefore, statement is true for n is equal to k plus 1. Statement is true for n is equal to k plus 1 whenever it is true for n is equal to k. Hence the given statement d by dx of x is to power n is equal to n x n minus 1 is true for all n. Hence proof for all positive integers n. Hence proofed. I hope the question is clear to you. Bye and have a nice day.