Differentiating Constant Functions

Remember that a constant function $ y(x) = c $ has the same value $ c $ at every point. The graph of such a function is a horizontal line:

Now at any point, the tangent line to the graph (remember this is the line which best approximates the graph) is the same horizontal line. Since the derivative measures the slope of the tangent line and a horizontal line has slope zero, we expect the following:

Derivative of a constant

$y^\prime(x) = \frac{dy}{dx}(x) = 0 \mbox{ if } y(x) = c.$

In fact, 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

$\frac{ y(x_0 + h) - y(x_0)}{h} = 0$

since $ y(x_0 + h) = y(x_0) = c
$ for a constant function.

Since the average rate of change is always zero, when we consider very small values of $ h $ , we see that the derivative $y^\prime(x) = 0$ for every value of $ x $ .