Logarithmic functions come in different flavors and bases. There's vanilla, strawberry...sugar cone, dish...but we'll start with the most basic one:
f(x) = ln x.
This function is defined for x > 0, and looks like this:
Think about the slope/derivative of this function. First off, since the function f(x) = ln x is always increasing, its derivative is always positive. Also, since f(x) = ln x is only defined for x > 0, f ' will also only be defined for x > 0. The graph of f ' will be entirely in the first quadrant:
If we take x = a close to zero, then the slope of f at a will be very steep:
Therefore f ' (a) will be large:
As x = a gets closer to zero, f ' (a) will be even larger:
If we take x = a far from zero, then the slope of f at a will be shallow, f ' (a) will be close to zero:
As x = a gets farther from zero, f ' (a) will move closer to zero:
If we fill in this rather sketchy graph of f ', we find the graph of 
for x > 0:
Here's the rule for finding the derivative of the natural log function:
If f(x) = ln x, then 
The graph is useful for remembering this rule. After we introduce the chain rule we'll see another way to find the derivative of ln x.