The derivative of a linear function mx + b can be derived using the definition of the derivative. The linear function derivative is a constant, and is equal to the slope of the linear function. Linear function derivatives are parts of many polynomial derivatives.
I want to talk about derivative of linear functions, so let's recall what a linear function is, a linear function is a function of the form f of x equals mx+b. Now the derivative is going to start with a definition of the derivative. So f prime of x equals the limit as h approaches zero of f of x plus h minus f of x over h. And I usually begin finding the derivative by looking at the difference quotient, so let's find and simplify the difference quotient, now in this case our f of x is mx+b. So f of x plus h is going to be m times x+h+b, m times the quantity x+h+b and then I subtract from that f of x so that's minus mx+b and I divide that by h. So let's distribute the m I get mx+mh, oops mh plus b minus and I have to distribute this minus sign over of both of these terms so minus mx and minus b all of that over h.
Now take a look we've got some cancellation here the mx cancels and the b's cancel. And so we're left with mh over h and even you get cancellation there, the h is canceled leaving just m so the whole difference quotient f of x plus h minus f of x over h simplifies to m. So this limit becomes the limit as h approaches zero of the constant m and that's just m and that kind of makes sense. That the derivative of a linear function should just be m the slope of the line. Right because the derivative gives us the slope of a curve at any point and so the slope of a line at any point should be m. Now another way to say this relationship between the linear function that's derivative is that the derivative with respect to x of mx+b is m. This is the derivative of a linear function.