 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 cos x from the first principle. So let us see what is the first principle. According to first principle f dash x is equal to limit h approaching to 0 function at x plus h minus function at x divided by h. So let us start with the solution to this question. First of all fx given to us is cos x. So function at x plus h will be cos x plus h. Now we apply the first principle. So we get f dash x is equal to limit h approaching to 0 cos x plus h minus cos x divided by h. Because according to first principle f dash x is equal to limit h approaching to 0 function at x plus h is cos x plus h function at x is cos x divided by h. This is equal to limit h approaching to 0 minus 2 sin x plus x plus h divided by 2 into sin x plus h minus x divided by 2 and this whole divided by h. This can be further written as limit h approaching to 0 minus 2 sin x plus h by 2 because here we get 2x plus h by 2 2 divided by 2x plus 2 divided by h gives us this into sin of x gets cancelled with minus x we have h by 2 and this whole divided by h. Now we multiply the numerator and denominator by 1 by 2 we get limit h approaching to 0 minus 2 sin x plus h by 2 into sin h by 2. Now this 2 gets cancelled with this 2. So here we have just minus sin left and this divided by this we can write as h by 2. Now we can say this is equal to limit h approaching to 0 minus sin x plus h by 2 into limit h approaching to 0 sin h by 2 divided by h by 2. We know that limit h approaching to 0 sin h by 2 by h by 2 is 1 here if we put h equal to 0 so we get minus sin x into 1 that is equal to minus sin x and this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.