 Hi, and welcome to this session, I am Shashi and I am going to help you to solve the following question. Question is if y is equal to cos inverse x, find d square y upon dx square in terms of y alone. Let us start with the solution now, we are given y is equal to cos inverse x, so differentiating both sides with respect to x, we get dy upon dx is equal to minus 1 upon under root of 1 minus x square. Now, again differentiating both sides with respect to x, we get d square y upon dx square is equal to, here we will apply the quotient rule, so under root 1 minus x square multiplied by derivative of minus 1 minus minus 1 multiplied by derivative of under root of 1 minus x square upon under root 1 minus x square whole square. Now this is further equal to 0 minus minus 1 multiplied by 1 upon twice under root 1 minus x square multiplied by minus 2x upon 1 minus x square, we know the derivative of minus 1 is equal to 0, so this whole term will be equal to 0. Now derivative of under root 1 minus x square is equal to 1 upon twice under root 1 minus x square multiplied by minus 2x, these two and two will get cancelled and we get minus x upon 1 minus x square multiplied by under root 1 minus x square, so we can write d square y upon dx square is equal to minus x upon 1 minus x square multiplied by under root 1 minus x square. Now we have to find the second derivative in terms of y, now we know that y is equal to cos inverse x, this is given in the question, this implies x is equal to cos y, now let us name this expression as 1, now substituting this value of x in equation 1 we get d square y upon dx square is equal to minus cos y upon 1 minus cos square y multiplied by under root 1 minus cos square y which is further equal to minus cos y upon sin square y multiplied by under root of sin square y which is equal to minus cos y upon sin cube y, now we can write it as minus cos y upon sin y multiplied by sin square y, so it is equal to minus cos y multiplied by cos x square y, so the second derivative is equal to minus cos y cos x square y, so the required second derivative in terms of y is equal to minus cos y cos x square y, this completes the question, hope you enjoyed the session, goodbye.