Final answer:
The derivative of the function g(x)=ln(9x+1) is found using the chain rule and simplifies to 9/(9x+1).
Step-by-step explanation:
To find the derivative of the function g(x)=ln(9x+1), we use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In this case, the outer function is the natural logarithm ln(x) and the inner function is 9x + 1.
The derivative of the outer function ln(x) is 1/x. Thus, the derivative of ln(9x+1) with respect to 9x+1 is 1/(9x+1). Next, we find the derivative of the inner function, which is 9x + 1. The derivative of 9x with respect to x is 9, and the derivative of a constant is 0, so the derivative of the inner function is 9.
By applying the chain rule, the derivative of g(x) is (1/(9x+1)) × 9, which simplifies to 9/(9x+1).