Differentiating Linear Functions

Here, we will find the derivative of the function $ y(x) = x.
$ Of course, the graph of this function is a line through the origin with slope one.

As for constant functions, the tangent line to the graph is the same line. Since the derivative measures the slope of the tangent line, we expect the following:

Derivative of the identity function

$\frac{dy}{dx} = 1 \mbox{ if } y(x) = x.$

Again, we can see this by computing the average change of the function. If we start with a point $ x_0
$ and compute the rate of change to some other point $ x_0 + h,
$ we see that

$\frac{ y(x_0 + h) - y(x_0)}{h} = \frac{(x_0 + h) - x_0}{h} = 1.$

Since the average rate of change is always one, when we consider very small values of $ h $ , we obtain $\frac{dy}{dx} = 1$ as we expected.