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.
Let f(z) = ln z. Find f'(z). |
Let f(x) = ln 4. Find f'(x). |
Find the derivative of each function. Make sure to use the letters given in the problem for the function and variable names.
Find the derivative of each function. Make sure to use the letters given in the problem for the function and variable names.
Find the derivative of each function. Make sure to use the letters given in the problem for the function and variable names.
Find the derivative of each function. Make sure to use the letters given in the problem for the function and variable names.
Find the derivative of each function. Make sure to use the letters given in the problem for the function and variable names.