 Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says, find the derivative of x raised to power n plus a into x raised to power n minus 1 plus a square into x raised to power n minus 2 and so on till a raised to power n minus 1 into x plus a raised to power n for some fixed real number a. Before starting with the solution, let us see the key idea behind the question that is the theorem 7 of your book that says that let fx be equal to a n x raised to power n plus a n minus 1 x raised to power n minus 1 and so on till a1x plus a0 be a polynomial function where a i's are all real numbers and a n is not equal to 0 then derivative function is given by dfx by dx is equal to n into a n into x raised to power n minus 1 plus n minus 1 into a n minus 1 into x raised to power n minus 2 and so on till 2a2x plus a1. Now let us start with the solution to this question. First of all, let fx be x raised to power n plus a into x raised to power n minus 1 plus a square into x raised to power n minus 2 and so on till a raised to power n minus 1x plus a raised to power n. Now we have to find f dash x that is the derivative of fx that is same as dfx by dx that will be equal to n into x raised to power n minus 1 plus n minus 1 into a into x raised to power n minus 2 plus a square into n minus 2 into x raised to power n minus 3 and so on till a raised to power n minus 1 into 1 plus 0. Now this happens because we see that using this theorem we get this because derivative of x raised to power n is n into x raised to power n minus 1. Now a a square a raised to power n minus 1 and a n they being constant they remain as it is in the respective terms x raised to power n becomes n into x raised to power n minus 1 x raised to power n minus 1 becomes n minus 1 into x raised to power n minus 1 minus 1 that is n minus 2. So we see that derivative of x raised to power n with respect to x is n into x raised to power n minus 1 and since the last term is a raised to power n that is a constant so we get a 0 here. So our answer to the question is n into x raised to power n minus 1 plus a into n minus 1 into x raised to power n minus 2 plus a square into n minus 2 into x raised to power n minus 3 and so on till a raised to power n minus 1. So this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.