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