Calculate the derivative of the function y = ln(x).

The derivative of y = ln(x) is 1/x.

To find the derivative of y = ln(x), we use the formula for the derivative of a natural logarithm function:

d/dx ln(x) = 1/x

This formula tells us that the derivative of ln(x) is equal to 1 divided by x. We can prove this using the definition of the derivative:

lim h->0 (ln(x+h) - ln(x))/h

= lim h->0 (ln((x+h)/x))/h

= lim h->0 (ln(1 + h/x))/h

= lim h->0 (1/x) (ln(1 + h/x)/h)

= (1/x) lim h->0 (ln(1 + h/x)/h)

We can evaluate the limit inside the parentheses using L'Hopital's rule:

lim h->0 (ln(1 + h/x)/h)

= lim h->0 (1/(1 + h/x)) (1/x)

= 1/x

Therefore, the derivative of y = ln(x) is 1/x.