find the derivative of y=ln3x with respect to x

Question

i need to know the steps and how its done to solve another problems like y=3overx

Answer 1

Question: find the derivative of y=ln3x with respect to x.

Use the chain rule.

The chain rule

If y = f(u) and u = g(x), then

dy/dx = (dy/du).(du/dx)

We have y = ln(3x)

Let u = 3x, then du/dx = 3

Now y = ln(u), with dy/du = 1/u, so

dy/dx = (dy/du).(du/dx)

dy/dx = (1/u).(3)= 3/u = 3/3x = 1/x

Answer: dy/dx = 1/x

Also,

y = ln(3x) = ln(3) + ln(x)  (from law of logarithms)

then

dy/dx = 0 + 1/x  (when you differentiate a constant value, it is zero)

dy/dx = 1/x

 

Answer 2

y=ln3x

dy/dx=(d/dx)(ln3x)

          =(1/3x)(3)

=1/x