 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 the following function that is cosecant x. So let us start with the solution to this question. First of all we see that according to first principle f dash x is equal to limit h approaching to zero function at x plus h minus function at x divided by h. So in this question we have to find the derivative of cosecant x using the first principle. In order to find out f dash x we have f x that is equal to cosecant x that is same as 1 upon sin x f dash x. In order to find out f dash x we have to find function at x plus h. So function at x plus h is equal to 1 upon sin x plus h. This we get by simply putting x equal to x plus h. Therefore using this principle we can say that f dash x will be equal to limit h approaching to zero. Function at x plus h is 1 upon sin x plus h minus function at x is 1 upon sin x divided by h. Now we see that this can be written as limit h approaching to zero. We have to subtract this from this. So we take the LCM of the denominators that is sin x into sin x plus h. In the numerator we will have sin x minus sin of x plus h divided by h. Now we have to simplify this. This is equal to limit h approaching to zero. Now sin x minus sin of x plus h can be written as 2 cos x plus x plus h by 2 sin x minus x minus h by 2 divided by h into sin x into sin of x plus h. Now this happens because we see that sin a minus sin b can be written as 2 cos a plus b by 2 into sin a minus b by 2. So we have got this. This can be written as limit h approaching to zero 2 into cos of, now x plus x is 2x. 2x plus h the whole divided by 2 becomes x plus h by 2 into sin. Now plus x gets cancelled with minus x. We have sin minus h by 2 and the whole divided by sin x into sin x plus h into h. Now we see that sin of minus h by 2 can be written as minus sin h by 2. Also we multiply the numerator and denominator by 1 by 2 and we get limit h approaching to zero minus 2 cos x plus h by 2 into sin h by 2. Now this 2 gets cancelled with this 2. So we have just minus sin left here and this divided by sin x into sin x plus h into h by 2. Now applying the limits we have limit h approaching to zero minus cos x plus h by 2 divided by sin x into sin of x plus h into limit h approaching to zero sin of h by 2 divided by h by 2. We know that limit h approaching to zero sin of h by 2 upon h by 2 is 1 and when here we put h equal to zero that is we apply the limits we get minus cos x divided by sin x into sin x multiplied by 1 and this is equal to minus cosecant x cot x because cos x by sin x can be written as cot x and 1 by sin x is cosecant x. So our answer to this question is minus cosecant x cot x. So I hope that you understood the question and enjoyed the session. Have a good day.