 Hi and welcome to the session. Let's work out the following question. The question says, find the derivative of cos square x with respect to x from first principle. Let us see the solution to this question. First of all, let y be equal to cos square x, we call this 1. Let delta x be a small increment in x, then delta y will be, this is delta x, then delta y will be small increment in the value of y, then we will have y plus delta y is equal to cos square x plus delta x and we call this 2. Now subtracting 1 from 2, we get del y is equal to cos square x plus del x minus cos square x. Now applying the formula of a square minus b square is equal to a plus b into a minus b, we have cos x plus del x plus cos x multiplied by cos x plus del x minus cos x, this can be written as cos x plus del x plus cos x into 2 into sin x plus del x by 2 into sin minus del x by 2. Now dividing both the sides by del x, we have del y by del x is equal to, now we see that sin of minus theta is equal to minus sin theta, so we can now write this as, this remains as it is that is cos x plus del x plus cos x into sin x plus del x by 2 into sin del x by 2 and the whole divided by del x by 2. Now taking the limit del x approaching to 0, we have dy by dx is equal to, because we see limit del x approaching to 0 del y by del x is equal to dy by dx, this is equal to, here we see that we will have a negative sign here, so we have minus cos x because we have taken del x to be 0 plus cos x into sin x plus 0 into 1, because we see that limit del x approaching to 0, sin del x by 2 divided by del x by 2 is equal to 1, this is further equal to minus 2 cos x sin x, that can also be written as minus sin 2 x, so this is our answer to this question, I hope that you understood the solution and enjoyed the session, have a good day.