Final answer:
The second derivative of the natural logarithm of x, ln(x), is found by first determining the first derivative, which is 1/x, and then differentiating again to get -1/x^2.
Step-by-step explanation:
To find the second derivative of the natural logarithm of x, ln(x), we first need to know the first derivative. The first derivative of ln(x) is 1/x, as it's a well-known result of differentiation. Applying the derivative operation again to find the second derivative, we will differentiate 1/x.
The derivative of 1/x with respect to x is -1/x^2. This negative comes from the rule that the derivative of x-1 (which is another way to write 1/x) is -1 times x to the power of negative two.
Therefore, the second derivative of the natural logarithm of x, ln(x), is -1/x^2.