What is the derivative of

Question

(x/3^x)?

Answer 1

d/dx(x / 3^x) = ( 3^x(d/dx(x)) - x d/dx(3^x) ) / (3^x)^2

This is in the form d/dx (u/v) which is equal to (vdu/dx - udv/dx) / v^2

                   = ( 3^x(1) - x (3^xlog3) ) / (3^x)^2

                   = 3^x [1 - x log3]  / (3^x)^2                       [Since d/dx(x) = 1, d/dx(a^x) = a^xloga]

                   = [1 - x log3]  / (3^x)

                   = (1 -  log3^x)  / (3^x)                                [Since nloga = loga^n]

Therefore d/dx(x / 3^x) = (1 -  log3^x)  / (3^x)