To find the derivative of 1, we will start by graphing the equation $y = 1$, pictured below:
The derivative of a function at a point is the rate of change of a function at said specific point. In other words, it is the slope of the tangent line to the curve at a given point on the function. For a straight line, it is just the slope of the line, since the tangent line to the curve is the line itself. We wish to find the derivative of
$$f(x) = 1$$
This means that we want to find the slope of the line $y = 1$, which is $\boxed{0}$, since $y =1$ is a horizontal line. This can be seen from the graph above.
This statement generalizes to any constant. The same logic applies to functions $f(x) = 3$, $f(x) = \pi$ or $f(x) = -\sqrt{5}$. The derivative of a constant function is always 0, regardless of the value of the constant. We could use the power rule, for example, to prove this result, but it is much more intuitive to provide a visual explanation. The visual explanation is much more important, since it provides the insight into why the derivative of any constant is 0 and you don’t have to take it for granted just because someone told you that there is a formula. We hope that this additional insight will help you remember why the derivative of any constant function is 0.
As an aside, remember that constants aren’t just the usual numbers like $7$ and $-31$. Expressions like $5+\sqrt3$ and $e^3 + \pi$ are also constants because they don’t have any variables.
