I'm learning to approximate functions using derivatives. I'm a little confused about the notation(s) $ dx $ (or $ dy $) and $ \Delta x $ (or $ \Delta y $). Do they mean the exact same thing -- except for the different curves? In other words, does $ dx $ represent the (small) increase in $ x $ for the straight line given in the image and $ \Delta x $ represent the (small) incrase in $ x $ for the given curve?
Do we use $ dx $ & $ \Delta x $, and $ dy $ & $\Delta y$ just to differentiate between the increments in the two different function or are they different?
Why is $ dx $ called the "differential" of $ x $ and $ \Delta x $ being called the "increment" of $ x $ (similarly for y)? My textbook also states:
We may note that the differential of the dependent variable is not equal to the increment of the variable where as the differential of independent variable is equal to the increment of the variable