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The Definition of the Derivative - Concept
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
The definition of the derivative is the slope of a line that lies tangent to the curve at the specific point. The limit of the instantaneous rate of change of the function as the time between measurements decreases to zero is an alternate derivative definition. The derivative is a function, and derivatives of many kinds of functions can be found, including linear, power, polynomial, exponential, and logarithmic functions.
I want to talk about the definition of the derivative. Now the definition of the derivative is related to the topics of average rate of change and the instantaneous rate of change. An average rate of change is really fundamental to the idea of derivative, let's start average rate of change, we call it average rate of change of a function is the slope of the secant line drawn between two points on the function. And the slope of secant line like any other slope of a line is going to be rise over run. Now if you notice I got these two points labeled with coordinates the rise is going to be f of a plus h minus f of a. And the run is going to be a+h-a, which is just h. And that's why the slope of the secant line if f of a plus h-f of a over h.
This is the slope of secant line which is the average rate of change of the function. Now instantaneous rate of change is what happens when we take the average rate of change over shorter and shorter increments of time. So we're letting h go to zero and as we do the secant line gets closer and closer to the tangent line, that's what this is. Now the way we do that is the way we get h to go to zero is we take limits. So the limit as h goes to zero of the average rate of change of f of a plus h minus f of a over h and taking the limit of that average rate of change is what gives us the instantaneous rate of change. This quantity is so important to Calculus it's given a much simpler symbol f prime of a this is the derivative of the function f at a and this symbol means the limit is h approaches zero of f of a plus h minus f of a over h. This concept is central to all of differential Calculus which is half of what we're going to do in this course.
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