In the previous exercise, we discussed the idea of the derivative at a point as the instantaneous rate of change at that point. The idea behind the definition is that by finding how the function changes over increasingly small intervals starting at a point, we can understand how a function changes exactly at that point. The formal expression of this concept is written as:

`$\lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$`

An equivalent interpretation is that the derivative is the slope of the tangent line at a point. A *tangent line* at a point is the line that touches the function at that point. Just as the slope of a line describes how a line changes, the slope of the tangent line describes how a “curvier” function changes at a specific point. The visual below shows what a tangent line looks like:
In the applet to the right, you can play around with the tangent line and see how it changes as you move to different values of *x*.

Our discussion of taking increasingly smaller intervals in the last exercise gives us a way to calculate the derivative of a function at a point, but this would be a laborious process if we always had to find the derivative at a point this way.

Fortunately, many common functions have corresponding derivative functions. Derivative functions are often denoted *f’(x)* (read *f* prime of *x*) or *df/dx* (read derivative of *f* at *x*). If f(x) represents the function value at a point *x*, the corresponding derivative function *f’(x)* represents how a function is changing at the same point *x*. Essentially, it saves us from having to draw a tangent line or take a limit to compute the derivative at a point. In later exercises, we will see a number of different rules at our disposal that allow us to calculate the derivatives of common functions.