 Hello friends, let's work out the following problem. It says find the derivative of the following function. The given function is x upon sine to the power nx. To find the derivative of this function, we'll be using quotient rule. It says d by dx of u by v is given by v into du by dx minus u into dv by dx upon v square that is the square of the denominator. So this knowledge will work as key idea. Let's now move on to the solution. We have to find the derivative of x upon sine x to the power n. Now here we'll apply the quotient rule and here u is x and v is sine x to the power n. Supplying the quotient rule v into du by dx now v is sine x to the power n into d by dx of x minus u that is x into d by dx of sine x to the power n upon v square that is sine x to the power n to the power 2. Again this is equal to sine x to the power n into derivative of x with respect to x is 1 minus x into derivative of sine x to the power n is n into sine x to the power n minus 1 into d by dx of sine x upon sine x to the power n to the power 2. Again this is equal to sine x to the power n minus x into n sine to the power n minus 1 x and the derivative of sine x is cos x upon sine x to the power n to the power 2 which is equal to sine x to the power n minus x into n sine x to the power n minus 1 can be written as sine x to the power n into sine x to the power minus 1 into cos x upon sine x to the power n to the power 2 can be written as sine x to the power n into sine x to the power n this is again equal to sine x to the power n minus n x sine x to the power n sine x to the power minus 1 can be written as 1 upon sine x into cos x upon sine x to the power n into sine x to the power n now taking sine x to the power n common from the numerator we have and taking the LCM we have sine x minus n x cos x upon sine x to the power n into sine x to the power n now this gets cancelled and we have and here we have 1 into 1 upon sine x when we take the LCM we will have 1 upon sine x here so we have sine x minus n x cos x upon sine x to the power n plus 1 hence the derivative of the given function is sine x minus n x cos x upon sine to the power n plus 1 x and this completes the question by for now take care have a good day