SECTION 2.3: Interpretations of the Derivative

The derivative can be noted either as f ' (x) or delta y/ delta x

Leibniz took this to mean a small difference in y over a small difference x. 

dy/dx can also be regarded as the derivative of y with respect to x. It essentially defines a very very small change in y divided by an equally small change in x. 

It is important to note that the units of the derivative of a function are the units of the dependent variable (y) divided by the units of the independent variable (x). If the derivative of a function does not change rapidly near a point, then it is approximately equal to the change in the function when the independent variable increases by one unit. 

An example of an everyday use of the derivative:

Derivative of velocity = dv/dt = acceleration, the units of which are meters per second squared.

Derivatives can also be used to estimate values of functions, i.e., the change in value of dependent quantity with every unit increase in independent quantity. In other words, local linear approximation can be found by using delta y = f ' (x) * delta x. 

I found it difficult to understand exactly at which point, the value of the derivative would be most accurate. As all values calculated seem to be approximations, what defines their accuracy and to how many decimal places?

This section is useful in Economics because understanding the unit changes if quantities is very  important. Marginal costs or benefits are always important when making big decisions. For example, to understand whether hiring one additional person to the labour force increases a firms efficiency/ profits or not.