Explain, in two different ways, without using the rules of differentiation, why the derivative of the constant function $f(x)=3$ must be $f^{\text{'}}(x)=0.[$Hint: Think of the slope of the graph of a constant function, and also of the instantaneous rate of change of a function that stays constant.$]($Select all that apply.$)$

$f^{\text{'}}(x)=0$ stays constant. Thus the graph has no instantaneous rate of change, making it a single point.

The graph of $f(x)=3$ is a vertical line placed at $3$ units. The slope $($derivative$)$ of a vertical line is zero.

The graph of $f(x)=3$ is a horizontal line placed at $3$ units. The slope of a horizontal line is infinity, so the derivative is zero.

A function that stays constant will have a rate of change of zero, so its derivative $($instantaneous rate of change.$)$ will be zero.

$f(x)=3$ has a constant rate of change of $3.$ Thus the graph's instantaneous rate of change is equal to its average rate of change.

The graph of $f(x)=3$ is a vertical line placed at $3$ units. The slope of a vertical line is infinity, so the derivative is zero.

$f(x)=3$ will have a graph that is a horizontal line at height $3,$ and the slope $($the derivative$)$ of a horizontal line is zero.Explain, in two different ways, without using the rules of differentiation, why the derivative of the constant function $f(x)=3$ must be $f^{\text{'}}(x)=0.[$Hint: Think of the slope of the graph of a constant function, and also of the instantaneous rate of change of a function that stays constant.$]($Select all that apply.$)$

$f^{\text{'}}(x)=0$ stays constant. Thus the graph has no instantaneous rate of change, making it a single point.

The graph of $f(x)=3$ is a vertical line placed at $3$ units. The slope $($derivative$)$ of a vertical line is zero.

The graph of $f(x)=3$ is a horizontal line placed at $3$ units. The slope of a horizontal line is infinity, so the derivative is zero.

A function that stays constant will have a rate of change of zero, so its derivative $($instantaneous rate of change.$)$ will be zero.

$f(x)=3$ has a constant rate of change of $3.$ Thus the graph's instantaneous rate of change is equal to its average rate of change.

The graph of $f(x)=3$ is a vertical line placed at $3$ units. The slope of a vertical line is infinity, so the derivative is zero.

$f(x)=3$ will have a graph that is a horizontal line at height $3,$ and the slope $($the derivative$)$ of a horizontal line is zero.