Example 1:

Find the derivative of the function

f(x) = x³.

In this and the following examples we let x vary over the real numbers and Δx vary over the nonzero infinitesimals. Let us introduce the new variable y with the equation y = x³. We first find Δy/Δx.

Next we simplify the expression for Δy/Δx.

Then we take the standard part.

Therefore,

We have shown that the derivative of the function

f(x) = x³

is the function

f'(x) = 3x²

with the whole real line as domain, f (x) and f'(x) are shown in Figure 2.1.3.

Figure 2.1.3