 Hi and welcome to the session. Today we will learn about derivatives of implicit functions. First of all let us learn what are implicit functions. An equation of the form f of x comma y is equal to 0 in which y is not expressible directly in terms of x is known as implicit function of x and suppose we have equation x plus cos xy minus y is equal to 0. Now here we cannot express y in terms of x so that means y is an implicit function. Let's take one example. Here we are given the equation 5x plus 3y square is equal to cos y and we need to find dy by dx. So in this equation we will differentiate both the sides with respect to x so we get 5 into derivative of x with respect to x is 1 plus 3 into now derivative of y square will be 2y into dy by dx equal to derivative of cos y is minus sin y into derivative of y with respect to x that is dy by dx. So from this we get 6y into dy by dx plus into dy by dx is equal to minus 5 so this gives us dy by dx is equal to minus 5 upon 6y plus sin y. From this example it must be clear to you that how we differentiate implicit functions. So let's move on to our next topic derivatives of inverse trigonometric functions. As we know that inverse trigonometric functions continuous so here to find the derivatives of inverse trigonometric functions we will use chain rule. So let's see the derivatives of inverse trigonometric functions for the function sin inverse of x its derivative is 1 upon square root of 1 minus x square and the domain of f dash over here is the open interval from minus 1 to 1. The derivative of cos inverse x is minus 1 upon square root of 1 minus x square and here the domain of f dash is the open interval from minus 1 to 1. For the function tan inverse of x its derivative is 1 upon 1 plus x square and the domain of f dash is r that is the set of real numbers. For f of x equal to cosecant inverse x its derivative is minus 1 upon x into square root of x square minus 1 and the domain of f dash is open interval from minus infinity to minus 1 union open interval from 1 to infinity. For f of x equal to cosecant inverse x f dash of x is equal to 1 upon x into square root of x square minus 1 and domain of f dash is open interval from minus infinity to minus 1 union open interval from 1 to infinity. Lastly for f of x equal to cot inverse x f dash of x is minus 1 upon 1 plus x square and the domain of f dash is set of real numbers. So here we will take one example. Suppose we are given y equal to sin inverse of 2x and we need to find dy by dx. So dy by dx will be equal to derivative of sin inverse of 2x. This will be equal to 1 upon square root of 1 minus 2x square into derivative of 2x using chain rule and this will be equal to 1 upon square root of 1 minus 4x square into now derivative of 2x will be 2. So this is equal to 2 upon square root of 1 minus 4x square. With this we finished this session. Hope you must have enjoyed it. Goodbye and have a nice day.