AP Calculus AB and BC

Tangent Line Tracer: Watch the Derivative Appear

A derivative is the slope of the tangent line to a curve at each point. Slide a point along f(x), read how steep the tangent is there, and plot that steepness as a height. String those heights together and they trace a whole new curve, f'(x): the slope of f, recorded everywhere at once.

f(x) = x^2drag the point along the curve
f'(x)traced only where you have dragged
a
1.000
f(a)
1.000
f'(a)
2.000

The slope of the tangent line above IS the height of the dot below. Where the tangent runs flat, the dot crosses the zero line. Drag from end to end of the curve to trace as much of the derivative graph as the point can reach.

A derivative answers one question at every point on a graph: how steep is the curve right here? Pick a point on ff, draw the line that just grazes the curve there (the tangent line), and measure its slope. That slope is the value of f′f' at that point. Topic 2.2 states it in one line: the derivative of a function at a point is the slope of the line tangent to the graph at that point (CHA-2.C.1).

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

That limit hands back a single number for a single input. The move that gives calculus its reach is running it at every xx at once. Slide the point across ff, record the tangent slope at each location, and the recorded slopes stack into a graph of their own. The height of that new graph at any xx equals the steepness of ff at the same xx. The new graph is the function f′(x)f'(x), and reading it against ff is the whole skill of Topics 5.8 and 5.9.

Read the derivative off the shape, before any formula

You can sketch the sign of f′f' without differentiating anything. Where ff climbs, its tangent tilts up, so f′>0f' > 0 and the derivative curve sits above the axis. Where ff falls, the tangent tilts down, so f′<0f' < 0 and the curve drops below. Where ff levels off at a peak or a valley, the tangent is flat, so f′f' touches zero and switches sign. That is exactly Topic 5.3 (the sign of f′f' marks where ff increases and decreases) and Topic 5.2 (a zero of f′f' is a critical point, a candidate for a maximum or minimum).

Steepness carries size as well as sign. A near-vertical stretch of ff throws f′f' far from the axis; a lazy, gently sloping stretch keeps f′f' hugging it. Watching the two graphs move in step turns the derivative from a rule you memorize into a shape you can predict before you compute a thing.

How to read it

The screen holds two linked panels. On the left is f(x)f(x) with a point you can grab and a tangent line pinned to it. On the right is a blank pair of axes that fills in as you move: every position of the point drops a dot at a height equal to the current tangent slope. Drag slowly and the right panel draws the graph of f′(x)f'(x) one dot at a time.

  • Drag onto a steep uphill section and watch the right-panel dot ride high above the axis; the steeper the climb, the higher it sits.
  • Drag onto a downhill section. The tangent tilts the other way and the dot falls below the axis into negative slope.
  • Ease the point onto a peak or a valley of ff. The tangent goes flat and the derivative dot lands right on the axis at f′(x)=0f'(x) = 0.
  • Sweep the point all the way across, let the full curve of f′f' appear, then check its shape against the prediction you made from ff alone.

The tell for a maximum

Watch the order of events at a hilltop of ff. Just before the peak the tangent still tilts up, so f′>0f' > 0; at the peak it is flat, so f′=0f' = 0; just after, it tilts down, so f′<0f' < 0. The derivative curve crossing from positive to negative is the graphical form of the First Derivative Test (Topic 5.4). A valley shows the mirror image: f′f' crosses from negative to positive.

If the tracer lets you switch functions, try one with a sharp corner such as f(x)=∣x∣f(x) = |x|. Approach the corner from the left and the right, and the tangent slope disagrees: no single value of f′f' exists there, so the right panel jumps rather than drawing a smooth point. That is Topic 2.4 in the open: a continuous graph can still fail to be differentiable, because the one-sided difference quotients do not meet (FUN-2.A.2).

Frequently asked questions

Is the derivative a number or a function?

Both, depending on what you ask for. f′(a)f'(a) at a single input is one number: the slope of the tangent at x=ax = a. Collect that slope at every input and you get f′(x)f'(x), a function in its own right (Topic 2.2). The tracer shows the switch happening in front of you: one dot is a number, the whole traced curve is the function.

Why does the derivative equal the slope of the tangent line?

The difference quotient shown above is the slope of a secant line through two points on ff. As h→0h \to 0 the second point slides into the first and the secant pivots into the tangent. The limit of those secant slopes is the tangent slope, which is exactly the definition of f′(x)f'(x) (Topics 2.1-2.2).

How do I sketch f′f' from a graph of ff without a formula?

Walk across ff from left to right and read its slope. Where ff rises, plot f′f' above the axis; where it falls, plot f′f' below; at every peak, valley, or flat spot, plot f′=0f' = 0. Steeper parts of ff push f′f' farther from the axis, gentler parts pull it closer. Mark the zeros first, then fill in between them. That is the graph-reading skill tested in Topic 5.8.