Final answer:
The derivative of ln(6x) is 1/x, the same as the derivative of ln(x), because the constant multiplier cancels out when applying the chain rule in differentiation.
Step-by-step explanation:
The derivative of ln(x) is 1/x. Now, let's consider the function ln(6x). By the chain rule of calculus, we would differentiate the outer function, which is the natural logarithm, and multiply it by the derivative of the inner function, which is 6x in this case.
The derivative of ln(u) with respect to u is 1/u, and the derivative of 6x with respect to x is 6, so we have:
d/dx [ln(6x)] = d/dx [ln(u)] * du/dx = 1/u * 6 = 6/(6x) = 1/x
This shows that the derivative of ln(6x) simplifies to 1/x, which is the same as the derivative of ln(x). It's critical to remember that the natural logarithm function's derivative is independent of any constant multiplier to the argument x, as the constants will cancel out due to the nature of the chain rule.