 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 sin x plus alpha upon cos x. To find the derivative of this function we will be using quotient rule. It says d by dx of u by v. The function u by v is given by v into du by dx minus u into dv by dx upon v square. This is called the quotient rule and this will be the key idea. Let us now move on to the solution. We have to find the derivative of sin x plus alpha upon cos x. So let u equal to sin x plus alpha and v be equal to cos x. Now we have to find the derivative of u by v using quotient rule is v square v into du by dx minus u into dv by dx. Let us now substitute the values of v and u in this. v is cos x into du by dx that is derivative of sin x plus alpha with respect to x minus u that is sin x plus alpha into dv by dx that is d by dx of cos x upon v square that is cos square x. Now this is equal to cos x into derivative of sin x plus alpha. Now since we are differentiating this with respect to x so here alpha is constant. So the derivative of sin x plus alpha will be cos alpha because derivative of sin theta is cos theta. But here theta is x plus alpha so it will be cos x plus alpha minus sin x plus alpha into derivative of cos x and we know that derivative of cos theta is minus sin theta so this becomes minus sin x because here in place of theta we have x upon cos square x. Now we will apply the formula of cos a plus b this is given by cos a cos b minus sin a sin b and we will also apply the formula of sin a plus b it is given by sin a cos b plus cos a sin b. Applying these formulas this becomes cos x and cos x plus alpha becomes cos x cos alpha minus sin x sin alpha because in place of a we have x and in place of b we have alpha minus it becomes plus minus of minus sin x is plus sin x and here we apply the formula of sin a plus b it becomes sin x cos alpha plus cos x sin alpha into sin x upon cos square x. Now multiplying cos x with this expression it becomes cos square x into cos alpha minus cos x sin x sin alpha plus sin x into sin x cos alpha is sin square x cos alpha and sin x into cos x into sin alpha is sin x cos x sin alpha upon cos square alpha it is cos square x. Now minus sin x cos x sin alpha gets cancelled with plus sin x cos x sin alpha. Now taking cos alpha from these two terms as common so it becomes cos alpha into cos square x plus sin square x upon cos square x. Now we know that cos square x plus sin square x is one so we have cos alpha into one upon cos square x which is equal to cos alpha upon cos square x hence the derivative of the given function is cos alpha upon cos square x and this completes the question bye for now take care have a good day