Derivative Builder
Slide a point along f and read the slope of the tangent there. Each
slope is dropped onto the second graph as a height. Sweep all the way across
and the derivative draws itself — f ′ is not a new kind of object, it
is a graph of slopes.
f (x ) =
Window
−2 to 2
−3 to 3
−2π to 2π
Try x^3-3x, sin(x), x^2,
abs(x), e^x, ln(x).
x^2
x^3 - 3x
sin(x)
e^x
abs(x)
Top: f , with the tangent at the point.
Bottom: every slope you have visited, plotted as a height.
The dashed line marks the same x on both , and
the red number
is written twice on purpose — slope up there, height down
here, the same f ′ and the same number. Drag on
either graph to move it.
f ″ you tracedf ′, as a height on f ″
Derivative Builder
The top graph shows f . A point slides along it, carrying a tangent
line. The slope of that tangent is plotted on the bottom graph as a
height . Sweep across and the bottom graph fills in: that is
f ′.
The slope, written twice
The same number is written on both graphs at once — beside the
tangent on f as slope = f ′ = … , and beside a
short red segment on f ′ as height = f ′ = … . The
grey word says what the quantity is called on the graph you are looking at;
the red
It is written as an equation rather than run together, because
"slope f ′" can be read as the slope of f′ — which is
f ″, and confusing those two is a live error in this course. The
equals sign forbids that reading.
Red means the slope at the point you are on , in both of its
guises — the tangent embodies it above, the segment measures it below.
Orange is kept for f ′ as a function : the dots you have
traced and the true curve when you reveal it.
?unitrun=true adds a slope triangle with a run of exactly 1,
so the rise is the slope. It is drawn only when that rise fits the
window: a steep slope against a shallow y -range would send it off
the frame, and it is left out rather than drawn clipped and lying about its
height.
The secant gap h (off by default)
Secant-to-tangent is a different question, and it has two tools of its own:
Secant to Tangent for the picture and
Derivative from the Definition
for the algebra. Here it competed with the tangent for the same line through
the same point, so it is hidden unless you ask for it with
?secant=true.
The dashed secant joins h toward 0 and
watch the secant settle onto the tangent, and the difference quotient
settle onto f ′(x ). That limit is the definition.
What to watch for
Where f is rising , the traced dots land above the
axis. Where f is falling , they land below .
At a peak or valley of f , the tangent is level and the dot
lands exactly on the axis. Students often expect a peak of
f to give a peak of f ′; it gives a zero.
The steeper f is, the further from the axis the dot lands —
in either direction.
Try |x| . The trace has a hole at x = 0, because there is no
single tangent line there. Not every function has a derivative
everywhere.
Keyboard
← → Move the point (focus the x slider first)
Space Start or stop the sweep
? This dialog
Esc Close
Close