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Derivatives of Linear Functions - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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