 Hello and welcome to the session my name is Mansi and I am going to help you with the following question. The question says for the function fx equal to x raise to power 100 by 100 plus x raise to power 99 by 99 and so on till x square by 2 plus x plus 100 plus x plus 1 prove that f dash 1 is equal to 100 times f dash 0 that means we have to prove that derivative of fx at x equal to 1 is equal to 100 times derivative of fx at x equal to 0. 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 an into x raise to power n plus an minus 1 into x raise to power n minus 1 and so on till a1x plus a0 be a polynomial function where a is are all real numbers and an is not equal to 0 then derivative function is given by dfx by dx is equal to n into an into x raise to power n minus 1 plus n minus 1 into an minus 1 into x raise to power n minus 2 and so on till 2a2x plus a1 so with this key idea let us start with the solution to the question now it is given to us that fx is equal to x raise to power 100 by 100 plus x raise to power 99 by 99 and so on till x square by 2 plus x plus 1 first of all we find out the derivative of fx that is f dash x that is same as tfx by dx according to the key idea this will be equal to 100 into x raise to power 99 divided by 100 plus 99 into x raise to power 98 divided by 99 and so on till x 2x by 2 plus 1 plus 0 now we see that here 100 gets cancelled with 199 with 99 2 with 2 so we have x raise to power 99 plus x raise to power 98 and so on till x plus 1 now f dash x at x equal to 1 we simply put in the value of x to be equal to 1 that will be 1 raise to power 99 plus 1 raise to power 98 and so on till 1 plus 1 since if we raise 1 to any power or any power raise to power 1 is 1 only so we have 100 times 1 so this is 1 1 and so on 100 times 100 times 1 is 100 now we find out f dash x at x equal to 0 this was f dash 1 we find out f dash 0 we simply put in the value of x to be equal to 0 so that is 0 plus 0 and so on till 0 plus 1 that will be equal to 1 since 100 is equal to 100 times 1 therefore we can say that dash 1 is equal to 100 times f dash 0 so this is what we were supposed to prove in this question I hope that you understood the question and enjoyed the session have a good day