# What is the instantaneous rate of change of #f(x)=3x+5# at #x_0=1#?

##### 1 Answer

For a linear function **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

One way to prove this is to imagine what happens when we change

Another is by utilizing the **power rule** for functions such as **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

#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#