(Using the Quotient Rule) -> Show that the derivative of (cosx)/(sinx) is (-1)/(sinx).

This question is a typical example aimed to test the student's understanding of the quotient rule, a technique which is used very often in calculus problems. Answer: For a function f(x) = cosx/sinx = u/v, let u = cosx and v =sinx Now, du/dx = -sinx and dv/dx = cosx d/dx (f(x)) = ( v du/dx - u dv\dx ) \ v^2 <- Quotient rule Applying the quotient rule: d/dx (cosx/sinx) = sinx(-sinx) - cosx(cosx) / sin^2(x) = -sin^2(x) - cos^2(x) / sin^2(x) = -1(sin^2(x) + cos^2(x)) / sin^2(x) (Using the fact: sin^2(x) + cos^2(x) = 1) = -1 / sin^2(x) as required. Method: > First assign values to u and v. > Then differentiate u and v. > Apply the quotient rule. > Simplify expression using trigonometric identity.