Definition of the Derivative
You've calculated your first derivative! Congrats. Just to summarize:
Definition: The Derivative of a function f(x) is another function, usually called f'(x), that describes the slopes of the tangent lines of f(x). It is traditional to pronounce "f'(x)" as "f prime of x"; in other words the ' is pronounced "prime."
Apology: I want to apologize about the very subtle notation (') of this incredible important concept. It's always struck me as ironic that one of the most important ideas in all of calculus gets such an insignificant tick mark to indicate it. I'm sorry, but there's not much I can do about it. There's a few alternate notations we can use, but for now, let's just stick with the annoyingly-teensy ' mark
Another way of saying the definition: The derivative f'(x) returns as an output the slope of the tangent line of f(x) at any input x.
You can see your first derivative from the previous activity below. Be sure you can connect the slope of the tangent line of f(x)=x^2+2x with the point moving along f'(x)=2x+2. Move A, and observe that the slope of the tangent line of f(x) at A is picked up by the derivative f'(x). You really need to take your time here and be sure you can make this connections.To help you connect the slope of the tangent line and the derivative, here's another applet illustrating the same concept as before, but with a different function,
f(x)=x^2-2x.
It turns out that the derivative is f'(x)=2x-2. You could find this out on your own by either studying the pattern in the slopes of the tangent lines (no need), or we'll see a huge shortcut in the coming lessons as well.Two things to ponder as you explore these applets:
- What about the slope of the tangent line tells you that
f(x)is at a minimum? - What happens to
f'(x)whenf(x)is at a minimum?