 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 2 divided by x plus 1 minus x square divided by 3x minus 1. So let us start with a solution to this question. First of all we see that derivative with respect to x of fx minus gx is equal to derivative with respect to x of fx minus derivative with respect to x of gx. If we put fx to be equal to 2 divided by x plus 1 and gx to be equal to x square by 3x minus 1 and putting these values here we get d by dx of 2 by x plus 1 minus x square by 3x minus 1 is equal to d by dx of 2 by x plus 1 minus d by dx of x square by 3x minus 1. So now we find derivative with respect to x of this function then we find the derivative with respect to x of this function then we subtract. So let us find first of all d by dx of 2 by x plus 1 now applying the quotient rule we will have d by dx of 2 upon x plus 1 is equal to first of all the denominator function that is x plus 1 into d by dx of the numerator function that is 2 minus 2 into d by dx of the denominator function that is x plus 1 divided by square of denominator this is equal to x plus 1 into we know that derivative of a constant is 0 so multiplied by 0 minus 2 into d by dx of x plus 1 is 1 because dx by dx is 1 and d by dx of 1 being a constant is 0 so we have 2 into 1 divided by x plus 1 the whole square this will be equal to minus 2 divided by x plus 1 the whole square now we have to find out d by dx of the second function that is x square by 3x minus 1 again we apply the quotient rule we get first of all in the numerator we have the denominator function into d by dx of numerator function that is x square minus numerator function into d by dx of denominator function that is 3x minus 1 this whole divided by square of the denominator this is equal to 3x minus 1 into 2x minus x square into d by dx of 3x minus 1 is 3 so x square into 3 because derivative with respect to x of 3x is 3 and derivative with respect to x of 1 being a constant is 0 so we have x square into 3 divided by 3x minus 1 the whole square let us open the bracket we get 3 2's are 6x square minus 2x minus 3x square divided by 3x minus 1 the whole square this can be written as 6x square minus 3x square is 3x square minus 2x divided by 3x minus 1 the whole square therefore our answer to the question is this minus this that is minus 2 divided by x plus 1 the whole square minus x into 3x minus 2 divided by 3x minus 1 the whole square so this is our answer to the question I hope that you understood the question and enjoyed the session have a good day