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Logarithmic Functions

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.

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