For a linear function y=mx+b, the rate of change, or derivative, is simply the slope m of the line. Thus, in this case the rate of change is 3 for every point along the function where the function is defined (in this case, for all real numbers x).
One way to prove this is to imagine what happens when we change x by one unit. Let us choose a generic point P, defined as (x_1, y_1). If we declare x_2 = x_1 +1, then our y_2 is equal to 3(x_2)+1 This is in turn equal to 3(x_1 + 1)+1 which is in turn equal to 3(x_1) +1 +3. From our initial equation, we see that this is equal to y_1 +3. Thus, for every increase of x by one unit, y increases 3 units.
Another is by utilizing the power rule for functions such as f(x)=x^n. The power rule tells us that for these functions, the derivative d/dx f(x) = nx^(n-1) Since a constant function c has a derivative with respect to x of 0, and the function f(x)=x is the same as x^1, we know that d/dx x = 1(x^0) = 1 Then from the constant multiple rule, we know that
d/dx (c*f(x)) = c*(d/dx (f(x)))
Using those equations and the Sum Rule, which states that
d/dx [f(x)+g(x)] = d/dx f(x) + d/dx g(x)
we can apply the power rule to a function of the sort y=mx+b, where b, the y-intercept, is a constant.
dy/dx = d/dx [mx +b]
= d/dx mx + d/dx b
= m*d/dx x + d/dx b
= m*1(x^0) + 0
= m
