 Hello and welcome to the session. In this session we discuss the following question which says find dy by dx if y is equal to sin inverse of x into square root of 1 minus x minus square root x into square root 1 minus x square the whole. Before we move on to the solution, let's discuss one formula which is sin of a minus b is equal to sin a into cos b minus cos a into sin b and we also have d by dx of sin inverse x is equal to 1 upon square root of 1 minus x square. This is the key idea that we use in this question. Let's move on to the solution now. We are given that y is equal to sin inverse of x into square root of 1 minus x minus square root x into square root of 1 minus x square the whole and we have to find dy by dx. First of all we suppose 3 equal to sin a and square root x is equal to sin b. Now using these values of x and square root x in this we get y is equal to and inverse of x where we put sin a into square root of 1 minus x and here in place of x we can put sin square b. Since we know that square root x is sin b so x would be sin square b. So here we have 1 minus sin square b minus square root x which is sin b into square root of 1 minus x square and in place of x we put sin a so 1 minus sin square a the whole. So further we have y is equal to sin inverse of into square root of 1 minus sin square b would be cos b since we know that sin square a plus cos square a is equal to 1 and from here we have cos square a is equal to square root of 1 minus sin square a minus sin b into cos a. Now for the key idea we have sin of a minus b is equal to sin a cos b minus cos a sin b. So using this in place of y is equal to sin inverse of sin of a minus b this means we have y is equal to a minus b. Now as we had assumed x to be equal to sin a so from here we can say that a is equal to sin inverse x so y is equal to sin inverse x minus b is assumed to be square root x so b would be equal to sin inverse of square root by dx is equal to b by dx of a inverse that is we have is equal to b by dx of minus b by dx of inverse square root x. Now we already know that b by dx of sin inverse x is equal to 1 upon square root of 1 minus x square so using this we get b by dx is equal to 1 upon 1 minus x square 1 upon square root of 1 minus square root x be whole square into b by dx of this gives us by dx is equal to 1 upon square root of 1 minus x square 1 upon square root of 1 minus x into b by dx of square root x would be 1 upon 2 into square root x. b by dx is equal to 1 upon square root of 1 minus x square minus 1 upon 2 into square root x into square root of 1 minus x c value of d y by dx so this is our final answer this complete c session hope you have understood the solution of this question.