Definition of Derivative
- Michael Hayashida
Our problem: trying to figure out the instantaneous ROC of f(x), graphed below, at x = 1. Let's get our bearings first:
- f(x) is in blue.
- The secant line between x = 1 and x = 1 + h is in brown. Note its slope is shown also.
- For some reason, the tangent line at x = 1 is shown also. It's in red. Note that you can toggle it on and off. You may want to start with it off so the drawing isn't too cluttered.
- The secant line's slope represents the __________ ROC of f(x) from ___ to ___.
- Do you agree that the smaller h is, the better the average ROC approximates the instantaneous ROC?
- Let's improve our approximation by shrinking h: drag the point labeled "1 + h" to the left.
- Note that as h approaches 0, the secant line starts becoming a __________ line at x = 1.
- Here is the gigantic realization of Newton (and some others as well): if the slope of the secant line is the average ROC of f(x), then the slope of the _________ line at x = 1 is the instantaneous ROC at x = 1.
- Write this slope using limit notation. Hint: h is approaching 0, and the formula for slope is - what?
If you wrote the slope of the tangent line properly, you've just written the definition of a derivative. It gets the bronze medal for the most important formula in calculus. You'll learn the gold and silver later. For now, when I ask you what the derivative of f(x) at x = 7 means, you should say, "It's the instantaneous ROC of f(x) at x = 7". Derivative means instantaneous ROC and vice versa. On a graph, the derivative at a point is the slope of the tangent line at that point. One last thing, the fraction part of the definition of derivative is called the difference quotient. It's vocab to know, nothing more.