Derivatives

A derivative is the point slope of a function.

You will recall from geometry that a slope is calculated from two points–how could a single point have a slope?

We do a limit calculation.

We get the slope of a point by doing the two point calculation and having a second point move closer and closer to the first point and the value that the calculation is approaching, the limit, is the answer for the point slope.

We show the two points for a slope cslculation below:

  • (x1, f(x1))
  • (x2, f(x2))

It is typical to change x1 to x and to change x+t where t is going to be a variable that gets smaller and smaller, approaching zero.

\dfrac{f(x2) - f(x1)}{x2 - x1}

With the change this becomes

\dfrac{f(x+t) - f(x)}{(x+t) - x}

The bottom simplifies to

\dfrac{f(x+t) - f(x)}{t}

We will use f(x)=x^2 as an example.

We want the derivative for the point (0, 0).

f(0)=0

0+t = t

\dfrac {f(x+t) - f(x)}{t} = \dfrac {f(t)}{t} = \dfrac {t^2}{t} = t

Since f(x)=t, as t approaches 0, f(x) approaches 0. Therefore the limit of f(x) as x approaches 0, is 0.