 Hi and welcome to the session and we pick up here. Let's discuss the question differentiate the following function with respect to x Log x raise to power x plus x raise to power log x. So, let's start the solution Let's assume Y is equal to Log x raise to power x plus x raise to power log x Again put U is equal to log x raise to power x and V is equal to x raise to power log x Then our Y will become is equal to U plus V Now differentiate both the sides with respect to x dy by dx is equal to du by dx plus dv by dx So, let us take this as number one. So, let us consider first U is equal to Log x raise to power x. So taking logarithm on both sides we get We get log U is equal to log U is equal to x log of log x Now differentiate both the sides with respect to x Using product rule. We have one by U Into du by dx is equal to x into derivative of log of log x plus log of log x into derivative of x that is 1 So this implies 1 by U into du by dx is equal to x Now derivative of log of log x. Now we know the derivative of log x is 1 over log x So derivative of log of log x is 1 over log x Into derivative of this log x. So this is 1 by x plus log of log x So this implies du by dx is equal to So here we have du by dx is equal to u into 1 over log x plus log of log x Substituting value of u here we get du by dx is equal to u as a log x raise to power x into 1 over log x plus log of log x Now consider x is equal to x raise to power log x Again method is same taking logarithm on both the sides log v is equal to log x log x into log x is equal to log x whose square is to power log x into 2 log x upon x. Now substituting value of du by dx and dv by dx In equation 1 we get dv by dv by dx is equal to du by dx plus dv by dx. Now du by dx is log x raise to power x into 1 over log x plus log of log x This is dv by dx is x raise to power log x into 2 log x upon So this is equal to dv by dx is equal to here take log x as a LCM so we get log x raise to power x minus 1 into 1 plus log x into log of log x plus x raise to power log x minus 1 into 2 log x So hence we have differentiate our given function and our answer is log x raise to power x minus 1 into 1 plus log x into log of log x is to power log x minus 1 into log x. I hope the question is clear to you. Bye and have a nice day