# Definition of Derivative

- Author:
- Michael Hayashida

- Topic:
- Calculus, Derivative

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.