 Hello friends let's work out the following problem it says find the derivative of the following function the given function is x to the power 4 into 5 sin x minus 3 cos x to find the derivative of this function will be using product rule it says dy dx of function u into v is u into dv by dx plus v into du by dx this will work as key idea let us now move on to the solution we have to find the derivative of x to the power 4 into 5 sin x minus 3 cos x now here we will use the product rule because this function is the product of two functions here u is x to the power 4 and d is 5 sin x minus 3 cos x so by the product rule it becomes the derivative is x to the power 4 into d by dx of 5 sin x minus 3 cos x plus v remains as it is that is 5 sin x minus 3 cos x into d by dx of x to the power 4 again this is equal to x to the power 4 into d by dx of 5 sin x minus 3 into d by dx of cos x now since 5 is a constant here also you can write it out of the bracket plus 5 sin x minus 3 cos x into derivative of x to the power 4 derivative of x to the power 4 is 4 into x to the power 3 as we know that derivative of x to the power n is n into x to the power n minus 1 here n is 4 now this is equal to x to the power 4 into 5 into d by dx of sin x derivative of sin x is cos x so this becomes 5 cos x minus 3 into derivative of cos x derivative of cos x is minus sin x plus 5 sin x minus 3 cos x into 4 x cube again this is equal to x to the power 4 into 5 cos x minus of minus sin x is plus sin x so this becomes plus 3 sin x plus 5 sin x minus 3 cos x into 4 x cube now taking x to the power 3 common from these two terms we have x to the power 3 into x into 5 cos x plus 3 sin x plus 4 into 5 sin x minus 3 cos x now this is equal to x to the power 3 multiplying x with this it becomes 5 x cos x plus 3 x sin x 4 into 5 sin x is 20 sin x 4 into minus 3 cos x is minus 12 cos x hence the derivative of the given function is x cube into 5 x cos x plus 3 x sin x plus 20 sin x minus 12 cos x and this completes the question bye for now take care have a good day