Describe the rate of change of f (x) = lnx. Your answer should explain how the slope changes when x is small and when x is large.

Explanation shown below

Step-by-step explanation:

f (x) = lnx;

The rate of change is defined as dy/dx;

dy/dx[Inx] = 1/x

and dy/dx is defined as the slope

The nature of the slope is as x increases; the slope decreases and conversely meaning as x decreases, the slope increases.

By plotting the graph of f (x) = lnx, you can conclude that when x is small, dy/dx has a larger value. For instance, the gradient of the curve when x=0.5 is 2. However, as you move along the x axis, you will see that the graph levels off, indicating a decrease in the slope, or dy/dx. For example, if x=10, dy/dx = 0.1 and when x=20, dy/dx = 0.05 and so on. Eventually, when x is large enough the value of dy/dx will be negligible.

Thus, as x increases, the slope decreases.