Final answer:
The derivative of a constant function is always 0, and the derivative of f(x) = mx is equal to the constant value m for all x.
Step-by-step explanation:
To prove that the derivative of a constant function is everywhere, we can use the geometric definition of the derivative. Let's say we have a constant function, f(x) = c, where c is a constant. The derivative of a function represents the slope of the tangent line at any given point. Since the function is constant, the tangent line will always be horizontal, and the slope of a horizontal line is always 0. Therefore, the derivative of a constant function is 0 for all x.
Now, let's consider the function f(x) = mx, where m is a constant. Using the same geometric definition of the derivative, we can find the slope of the tangent line at any point on the graph of this function. Since the function is a straight line with a constant slope, the derivative will also be a constant. The slope of the line is equal to m, so the derivative of f(x) = mx is f'(x) = m for all x.