 Hello friends welcome to the session I am Malka. I am going to help you find dy by dx n y equal to secant inverse 1 upon 2x square minus 1 where 0 is less than x is less than 1 by root 2. Now let us start with the solution y equal to secant inverse 1 upon 2x square minus 1. Now let x cos theta this implies theta equal to cos inverse x this is our first equation. Now this implies y equal to secant inverse into 1 upon cos 2 cos square theta minus 1 this implies y equal to secant inverse into 1 upon cos 2 theta as we all know that cos 2 theta is equal to 2 cos square theta minus 1. So this implies y equal to secant inverse into secant 2 theta this implies y equal to 2 theta this implies y equal to 2 into cos inverse x. Now differentiate both sides with respect to x we get dy by dx equal to 2 into dx of cos inverse x. This implies dy by dx equal to 2 into minus 1 upon square root of 1 minus x square. Therefore dy by dx equal to minus 2 upon square root of 1 minus x square. Hope you understood the solution and enjoyed the session. Goodbye and take care.