The Derivative

For six classes you have been sneaking up on an instant. Average speed over a minute, over a second, over a tenth of a second — each secant a little more honest than the last, and at the boards in How Fast Right Now? your group watched the numbers settle toward one value they never quite reached. The Limit gave that value a name. The derivative is what you get when you build the whole idea into one definition:

$f^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$

Read it slowly, because every piece is something you did with your hands: $\frac{f(a+h)-f(a)}{h}$ is the slope of a secant from $a$ to a nearby point; $h\to 0$ marches the nearby point home; the limit is the value the march settles on. The derivative of $f$ at $a$ is the slope of the tangent there — the speed right now, not over any interval at all.

One number, three costumes

Costume	What $f^{\prime }(a)$ is	Where you meet it
Geometric	Slope of the tangent at $(a,f(a))$	Curve Sketching
Physical	Instantaneous rate of change — velocity, growth, flow	Motion on a Line
Algebraic	The limit of difference quotients	Limits Practice

The costumes matter because problems arrive wearing them. A question about a tangent line, a question about a falling stone, and a question about a shrinking limit are the same question, and the mark of fluency in this course is hearing that.

From a point to a function

Compute $f^{\prime }(a)$ at enough points and a pattern appears — the slopes themselves form a function, $f^{\prime }(x)$, the derivative function. For $f(x)=x^2$ the definition gives, at any $x$:

$\begin{array}{rl}{f}^{\prime }(x)& =\underset{h\to 0}{\lim }\frac{(x+h{)}^2-x^2}{h}\\ & =\underset{h\to 0}{\lim }\frac{2xh+h^2}{h}\\ & =\underset{h\to 0}{\lim }(2x+h)=2x\end{array}$

• Check that against your group’s secant tables from Zooming In: at $x=3$, were the slopes settling near 6?
• Try the same three-line computation for $f(x)=x^3$.
• Conjecture what $f(x)=x^{17}$ will do — then bring your conjecture to class, where the toolbox starts from exactly this pattern.

One honest warning: the definition is the meaning, and the rules that follow are the shortcuts. When a problem confuses you later in the course, come back here — the definition never does.

Curriculum connection

A2.1

determine numerically and graphically the intervals over which the instantaneous rate of change is positive, negative, or zero for a function that is smooth over these intervals (e.g., by using graphing technology to examine the table of values and the slopes of tangents for a function whose equation is given; by examining a given graph), and describe the behaviour of the instantaneous rate of change at and between local maxima and minima Sample problem: Given a smooth function for which the slope of the tangent is always positive, explain how you know that the function is increasing. Give an example of such a function.

Link to original

A2.2

generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, $f(x)$, for various values of $x$ (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, $f^{\prime }(x)$ or $\frac{dy}{dx}$, and make connections between the graphs of $f(x)$ and $f^{\prime }(x)$ or $y$ and $\frac{dy}{dx}$ [e.g., when $f(x)$ is linear, $f^{\prime }(x)$ is constant; when $f(x)$ is quadratic, $f^{\prime }(x)$ is linear; when $f(x)$ is cubic, $f^{\prime }(x)$ is quadratic] Sample problem: Investigate, using patterning strategies and graphing technology, relationships between the equation of a polynomial function of degree no higher than 3 and the equation of its derivative.

Link to original