 Hello and welcome to the session my name is Mansi and I am going to help you with the following question. The question says find the derivative of the following function from first principle and that is x plus 1 divided by x minus 1. 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 0 function at x plus h minus function at x divided by h. So in order to find out f dash x we have to find f x that is given to us is x plus 1 divided by x minus 1 function at x plus h we can get by simply replacing x by x plus h that will be x plus h plus 1 divided by x plus h minus 1. So now we can say that f dash x is equal to limit h approaching to 0 function at x plus h is x plus h plus 1 divided by x plus h minus 1 minus function at x that is x plus 1 divided by x minus 1 the whole divided by h. Now let us simplify it further this is equal to limit h approaching to 0 x plus h minus 1 into x minus 1 now we are subtracting these two here we will have x plus h plus 1 into x minus 1 minus x plus h minus 1 into x plus 1 this divided by h this is equal to limit h approaching to 0. Now let us open the brackets we will have x square minus x plus h x minus h plus x minus 1 minus x square plus x plus h x plus h minus x minus 1 divided by h into x minus 1 into x plus h minus 1 now this is equal to limit h approaching to 0 now x square gets cancelled with minus x square. So we will have minus x minus x is minus 2 x plus h x will get cancelled with minus h x minus h minus h is minus 2 h plus x minus of minus x is plus or plus 2 x minus 1 plus 1 they get cancelled divided by h into x minus 1 into x plus h minus 1. Now we see that minus 2 x gets cancelled with plus 2 x and we have limit h approaching to 0 minus 2 h divided by h into x minus 1 into x plus h minus 1 we see that h gets cancelled with h and we have limit h approaching to 0 minus 2 divided by x minus 1 into x plus h minus 1 now applying the limit that is putting h equal to 0 we will have minus 2 divided by x minus 1 into x minus 1 that is equal to minus 2 by x minus 1 the whole square. So our answer to this question is minus 2 divided by x minus 1 the whole square. I hope that you understood the question and enjoyed the session. Have a good day.