 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, if y is equal to square of tan inverse x, show that x square plus 1 whole square multiplied by y2 plus 2x multiplied by x square plus 1 multiplied by y1 is equal to 2. Let us start the solution. We have given y is equal to square of tan inverse x. Now differentiating both sides with respect to x we get y upon dx is equal to 2 tan inverse x multiplied by derivative of tan inverse x. Here we have applied the chain rule to find derivative of tan inverse x square. Now this implies dy upon dx is equal to 2 tan inverse x multiplied by 1 upon 1 plus x square. We know derivative of tan inverse x is equal to 1 upon 1 plus x square. Now this is equal to 2 tan inverse x upon 1 plus x square. Now we can write first derivative of y as y1. So we can write dy by dx as y1 is equal to 2 tan inverse x upon 1 plus x square or we can say this implies 1 plus x square multiplied by y1 is equal to 2 tan inverse x. Let us name this expression as 1. Now we know dy upon dx is equal to 2 tan inverse x upon 1 plus x square. Now if we differentiate both the sides with respect to x again then we get d square y upon dx square is equal to here we will apply the quotient rule. We get 1 plus x square multiplied by derivative of 2 tan inverse x minus 2 tan inverse x multiplied by derivative of 1 plus x square 1 plus x square whole square. Now this implies d square y upon dx square is equal to 1 plus x square multiplied by 2 upon 1 plus x square. We know derivative of 2 tan inverse x is equal to 1 plus x square minus 2 tan inverse x multiplied by 2x. We know derivative of 1 plus x square is 2x upon 1 plus x square whole square. Now simplifying we get d square y upon dx square is equal to 2 minus mu x tan inverse x upon 1 plus x square whole square. Here 1 plus x square and 1 plus x square will cancel each other and we get 2 and we know 2x multiplied by 2 is equal to 4x. So, we get minus 4x multiplied by tan inverse x. Now multiplying both sides by 1 plus x square whole square we get 1 plus x square whole square d square y upon dx square is equal to 2 minus 4x tan inverse x. Now this implies 1 plus x square whole square d square y upon dx square plus 4x tan inverse x is equal to 2 adding 4x tan inverse x on both sides. We get this expression. Now this implies 1 plus x square whole square d square y upon dx square plus 2x multiplied by 2 tan inverse x is equal to 2. Now from expression 1 we know 2 tan inverse x is equal to 1 plus x square multiplied by y 1. Now let us name this expression as 2. Now substituting the value of 2 tan inverse x from expression 1 in expression 2 we get 1 plus x square whole square d square y upon dx square plus 2x multiplied by 1 plus x square multiplied by y 1 is equal to 2. We know 2 tan inverse x is equal to 1 plus x square multiplied by y 1. So, we get this expression. Now we can write second derivative of phi as phi 2. So, we get 1 plus x square whole square multiplied by y 2 plus 2x multiplied by 1 plus x square multiplied by y 1 is equal to 2. We can write 1 plus x square as x square plus 1. So, we get x square plus 1 whole square multiplied by y 2 plus 2x multiplied by x square plus 1 multiplied by y 1 is equal to 2. So, this is our required answer and it is proved. This completes the session. Hope you understood the session. Keep smiling and take care.