To find the derivative of 1, we will start by graphing out the equation $y = 1$, pictured below:

We define the derivative of a function at a specific point as 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. As a reminder, our function is:

$$f(x) = 1$$

If we look at this function, we can see that the value of the function is always 1, no matter what the value of $x$ is. Thus, if we select any value of $x$ and find the tangent line to it, then we see that the tangent line should have a slope of zero since there is no change along the entirety of $f(x)$. $f(x)$ is always $1$, so the slope of $f(x)$, or in other words the derivative of 1, is $0$.