 Hello and welcome to the session. Let us discuss the following question. It says find the derivative of the following function. The given function is 4x plus 5 sin x upon 3x plus 7 cos x. To find the derivative of this function we will be using quotient rule. It is d by dx of the function u by v is given by v into du by dx minus u into dv by dx upon v square. So this knowledge will work as key idea. Let us now move on to the solution. We have to find the derivative of 4x plus 5 sin x upon 3x plus 7 cos x. Now here we will apply the quotient rule. So here u is 4x plus 5 sin x and v is 3x plus 7 cos x. So applying the quotient rule we have 3x plus 7 cos x into d by dx of u that is 4x plus 5 sin x minus u that is 4x plus 5 sin x into d by dx of v that is 3x plus 7 cos x upon v square that is 3x plus 7 cos x whole square. Now this is equal to 3x plus 7 cos x into derivative of 4x plus 5 into d by dx of sin x minus 4x plus 5 sin x into d by dx of 3x plus d by dx of 7 cos x upon 3x plus 7 cos x whole square. Again this is equal to 3x plus 7 cos x. Now the derivative of 4x is 4 because derivative of x with respect to x is 1. So 4 into 1 is 4 plus 5 into derivative of sin x and the derivative of sin x is cos x so this becomes 5 cos x minus 4x plus 5 sin x into derivative of 3x is 3 plus derivative of 7 cos x is minus sin x because derivative of cos x is minus sin x so this becomes minus 7 sin x upon 3x plus 7 cos x whole square. 3x plus 7 cos x into 4 plus 5 cos x minus 4x plus 5 sin x into 3 minus 7 sin x upon 3x plus 7 cos x whole square. Now multiplying these two brackets we have 4x 3x into 4 is 12x 3x into 5 cos x is 15x cos x 7 cos x into 4 is 28 cos x 7 cos x into 5 cos x is 35 cos square x minus 4x into 3 is 12x 4x into minus 7 sin x is minus 28x sin x now 5 sin x into 3 is plus 15 sin x 5 sin x into minus 7 sin x is minus 35 sin square x upon 3x plus 7 cos x whole square now this is equal to 12x plus 15x cos x plus 28 cos x plus 35 cos square x opening the brackets sin will change becomes minus 12x plus 28x sin x minus 15 sin x plus 35 sin square x upon 3x plus 7 cos x whole square now plus 12x gets cancelled with minus 12x and taking 35 common from these two terms we have 35 into cos square x plus sin square x plus 15x cos x plus 28 cos x plus 28x sin x minus 15 sin x upon 3x plus 7 cos x whole square now we know that cos square x plus sin square x is equal to 1 so this becomes 35 into 1 is 35 so 35 plus 15x cos x plus 28 cos x plus 28x sin x minus 15 sin x upon 3x plus 7 cos x whole square so this is the derivative of the given function that is 35 plus 15x cos x plus 28 cos x plus 28x sin x minus 15 sin x upon 3x plus 7 cos x whole square and this completes the question bye for now take care have a good day