The rule for differentiating ln(x) is 1/x. This means that when you take the derivative of ln(x), you will end up with 1/x as the new coefficient.
When differentiating ln(x^3), you will first use the power rule to bring down the exponent of 3, giving you 3ln(x). Then, you can apply the rule for differentiating ln(x), which is 1/x. This will give you a final answer of 3/x.
The derivative of ln(ln(x)) is 1/(xln(x)). This can be found by applying the chain rule, and then using the rule for differentiating ln(x) twice.
Yes, you can differentiate ln(x^2 + 1). Using the chain rule, the derivative would be 2x/(x^2 + 1). This can also be written as 2x/(x^2 + 1)^2.
The derivative of ln(x^3 + 5x) is (3x^2 + 5)/(x^3 + 5x). This can be found by using the quotient rule, with the top part being the derivative of x^3 + 5x, and the bottom part being the original function.