The Function picker on the left lists seven families in two groups, Polynomial and Transcendental; tapping one loads f with its derivative and puts x0 back at 1. Learn more about the function families
The Display checkboxes f(x), f'(x) and tangent each hide or show one layer, and the legend under the plot lists only the layers that are on; Accent color under Appearance recolours the highlights. Learn more about the display toggles
The point x₀ slider under Parameters moves the probe from −5 to 5 in steps of 0.05; the tangent, both markers and the numbers follow it, and Reset returns x0 to 1. Learn more about the x0 slider
The plot draws f as a solid curve, f′ as a dashed curve and the tangent at x0 as a light-blue line, with a vertical link line through both curves at x0. Learn more about getting started
The Jump to bar under the plot groups special points as Roots of f, Extrema of f and Inflections; tapping a value puts x0 exactly there, and a family with none shows none in view. Learn more about the jump-to points of interest
The At the point x₀ card shows f(x₀), f'(x₀) and Tangent slope side by side; the last two are always the same number. Learn more about the at-the-point card
Tangent line — the straight line that just touches the curve of f at a single point, sharing the same slope as f there. Its slope is f′(x0).
f prime as a function — the derivativef′ is a function of x in its own right. Its value at any x is the slope of f at that same x.
Critical point — a value of x where f′(x)=0 or f′(x) is undefined. Local maxima and minima of f live at critical points.
Inflection point — a value of x where f changes concavity. These are extrema of f′ (where f′′ would be zero).
Closed-form derivative — an exact symbolic formula for f′, like dxd[x2]=2x or dxd[sinx]=cosx.
Getting Started
The page opens with the Quadratic family f(x)=x2 already loaded and the probing point x0 parked at 1. You see three things at once on the graph:
• The solid blue curve of f.
• The dashed deeper-blue curve of f′.
• A light-blue tangent line touching f at x0.
A faint vertical dashed line drops through both curves at x0, so you can read off f(x0) and f′(x0) on the same vertical slice. The boxed card below the graph shows both numbers plus the tangent slope — all three are pictures of the same derivative value.
To explore quickly, drag the point x₀ slider in the left panel, or click any of the Jump to buttons to snap onto roots, extrema, or inflection points.
The Function Families
DemoPicking a family
Step 0 of 5
Seven families are organized into two groups in the left panel.
Polynomial:
• Identity — f(x)=x, f′(x)=1. The constant derivative is the baseline case.
• Quadratic — f(x)=x2, f′(x)=2x. The derivative is the line through the origin with slope 2.
• Cubic — f(x)=x3, f′(x)=3x2. The derivative is a parabola; the origin is an inflection point of f.
Transcendental:
• Sine — f(x)=sinx, f′(x)=cosx. Two waves locked a quarter-period out of phase.
• Cosine — f(x)=cosx, f′(x)=−sinx. Same shape, shifted the other way.
• Exponential — f(x)=ex, f′(x)=ex. The function and its derivative are the same curve.
• Logarithm — f(x)=lnx, f′(x)=1/x. Only defined for x>0.
Identity: a Constant Derivative
f(x)=x is the straight line through the origin at 45 degrees, and its derivative is the constant f′(x)=1. On the graph that shows up as a horizontal dashed line sitting at height 1 — the same value no matter where x0 goes.
At x0=1 the readings are f(x0)=1 and f′(x0)=1. The tangent to f is the line itself, because a straight line is its own best linear approximation everywhere.
Identity, frozen at x0 = 1
f is the 45-degree line, f′ the flat dashed line at height 1. The tangent lies exactly on f, because a straight line is its own tangent everywhere.
Drag x0 anywhere and the tangent never tilts. That is what a constant derivative means: the rate of change does not depend on position.
The two curves are worth comparing as objects rather than as pictures. f is a line with slope 1; f′ is a line with slope 0. Differentiating dropped the degree by one, which is the pattern the quadratic and cubic families continue.
Quadratic: a Derivative That Grows Linearly
f(x)=x2 has f′(x)=2x — a parabola paired with a straight line through the origin.
At x0=1: f(x0)=1 and f′(x0)=2. The tangent on the parabola climbs at slope 2, and the dot on f′ sits at height 2. Those are the same number seen two ways, which is the whole point of showing both curves at once.
Quadratic, frozen at x0 = 1
The parabola with its derivative drawn as the line f′(x) = 2x. At x0 = 1 the tangent slope is 2 and the dashed line passes through height 2 - the same number in two places.
The sign structure is easy to read here. f′(x)=2x is negative for x<0, zero at the origin, positive for x>0 — so the parabola falls, levels off, then rises. The single zero of f′ marks the single turning point of f.
Notice also that f′ crosses zero rather than merely touching it. That crossing is what makes the origin a genuine minimum instead of a flat spot, a distinction the cubic makes vivid by failing it.
Cubic: a Flat Spot That Is Not an Extremum
f(x)=x3 has f′(x)=3x2, an upward parabola that touches zero at the origin and is positive everywhere else.
At x0=1: f(x0)=1 and f′(x0)=3. Slide x0 to the origin and the tangent goes flat — but the curve does not turn around.
Cubic, frozen at x0 = 1
f′(x) = 3x² sits entirely above the axis apart from a single touch at the origin. That touch without a crossing is why x³ has a flat spot but no turning point.
This is the standard counterexample to "zero derivative means extremum". At x=0 the tangent is horizontal, yet f′≥0 everywhere, so the function never stops increasing. It merely pauses.
What distinguishes this from the quadratic is that f′ touches the axis without crossing it. No sign change, no turning point. The rule that actually classifies a critical point is the sign change of f′, not the zero itself.
Sine: the Derivative Is a Quarter Turn
f(x)=sin(x) has f′(x)=cos(x) — the same wave, shifted left by π/2.
At x0=1: f(x0)=0.84 and f′(x0)=0.54. The sine is still climbing at that point, though it is slowing down as it nears its peak at π/2≈1.571.
Sine, frozen at x0 = 1
Two waves a quarter period apart. The marker on f′ reads 0.54 while f reads 0.84 - the sine is still rising, but slowing as it approaches its peak.
The shift is the most useful thing to see here. Every peak of sin lines up with a zero of cos going downward, every trough with a zero going upward, and the steepest parts of the sine sit exactly where the cosine reaches ±1.
Differentiating four times returns you to where you started: sin→cos→−sin→−cos→sin. Each step is another quarter turn, which is why trigonometric derivatives cycle rather than simplify.
Cosine: the Same Wave With a Sign
f(x)=cos(x) has f′(x)=−sin(x). Same shape as the sine family, but the minus sign matters.
At x0=1: f(x0)=0.54 and f′(x0)=−0.84. The cosine is descending there — it peaked at the origin and is on its way down to its zero at π/2.
Cosine, frozen at x0 = 1
The same pair with the roles swapped and a sign attached: f reads 0.54 and f′ reads -0.84. The dashed curve is below the axis wherever the cosine is falling.
Compare this with the sine family directly: the readings swap roles and one of them picks up a minus. That is not a coincidence but the quarter-turn rule applied one step further along the cycle.
The negative sign is also the reason the second derivative of cos is −cos — the function equals the negative of its own second derivative, which is exactly the differential equation that makes sine and cosine describe oscillation.
Exponential: Its Own Derivative
f(x)=ex has f′(x)=ex. The two curves coincide exactly, so the solid and dashed lines lie on top of each other across the whole graph.
At x0=1: f(x0)=2.72 and f′(x0)=2.72. Both dots land at the same height, and they will do so wherever x0 goes.
Exponential, frozen at x0 = 1
Solid and dashed curves coincide across the whole frame, and both markers land at 2.72. This is the graph of a function equal to its own derivative.
This is the defining property of ex, not a curiosity of it. Any function satisfying f′=f has to be Cex for some constant C, and requiring f(0)=1 pins down C=1. That is one legitimate way to define the number e in the first place.
Geometrically it says the height of the curve is its own slope: at height 2.72 the tangent rises at 2.72 per unit. Growth proportional to current size is exactly what compound interest, population models and radioactive decay all express.
Logarithm: a Derivative That Blows Up
f(x)=ln(x) has f′(x)=1/x, and both are defined only for x>0 — the left half of the graph is empty for this family.
At x0=1: f(x0)=0 and f′(x0)=1. The logarithm crosses the x-axis at 1, and it does so at slope exactly 1.
Logarithm, frozen at x0 = 1
Nothing is drawn left of the y-axis: both ln x and 1/x are undefined there. At x0 = 1 the log crosses zero while its derivative reads exactly 1.
Watch what happens as x0 moves toward zero. The logarithm plunges without bound, and 1/x climbs without bound alongside it — the curve gets steeper and steeper with no limiting slope. Move the other way and both flatten out: at x=5 the slope is already down to 0.2.
That contrast is worth holding next to the exponential, which is the inverse function. Where ex grows faster than any polynomial, lnx grows slower than any positive power of x — and the reciprocal derivative is precisely why.
The x₀ Slider
DemoMoving x₀
Step 0 of 5
The point x₀ slider sweeps the probing point from −5 to +5 in steps of 0.05. As you drag:
• The tangent line moves along f, always touching at the new x0.
• The dashed vertical link line follows.
• The marker dot on f at (x0,f(x0)) slides along with it.
• The marker dot on f′ at (x0,f′(x0)) slides with it too — at the same height as the tangent slope.
• The three numeric cards below update live.
The whole point of the slider is to make the connection between the tangent slope and the height of f′ feel concrete. Pick a steep part of f — the tangent tilts hard, and f′ flies far from zero. Pick a flat part — the tangent goes horizontal, and f′ touches zero.
The Reset button next to the Parameters label returns x0 to the family's default starting position.
Jump-to Points of Interest
DemoJump to roots, extrema, inflections
Step 0 of 5
Below the graph is a row of buttons grouped into three categories — the three kinds of special points worth visiting:
• Roots of f — where f(x)=0. The curve of f crosses the x-axis here. For x2 that's only x=0; for sinx it's every multiple of π.
• Extrema of f — where f′(x)=0. The tangent goes horizontal; f has a local max or min. For x2 that's x=0 (the minimum at the origin).
• Inflections — where f′′(x)=0, equivalently extrema of f′. The concavity of f flips, and f′ is at its own local max or min. For x3 that's x=0.
Clicking any button snaps x0 to that exact location, so you can hop between landmarks without fighting the slider for a clean number.
Reading the At-The-Point Card
The boxed card below the graph displays the three quantities at x0 side by side:
• f(x₀) — the height of the function at x0. Where the function value lives.
• f'(x₀) — the height of the derivative curve at the same x0. The slope of f there.
• Tangent slope — the slope of the tangent line you see on the plot. Identical to f′(x0).
The middle and right entries are always the same number. That repetition is the whole pedagogical point: the derivative, the height of f′, and the slope of the tangent are three pictures of one quantity.
Skim across the row as you slide x0. The left number tells you what f is doing; the middle and right tell you, in stereo, how fast it's changing.
Display Toggles
DemoDisplay, plot size and tabs
Step 0 of 5
The Display section in the left panel lets you hide any of the three layers when one of them is in the way:
• f(x) — toggles the solid curve of the original function. Off, you see just f′ and the tangent, which is sometimes useful when the function curve is busy.
• f'(x) — toggles the dashed derivative curve. Off, the picture reduces to the classical tangent line at a point without the derivative-as-function overlay.
• tangent — toggles the tangent line at x0. Off, you see just the two function curves and their link line.
The toggles are independent — any combination is valid. The legend below the graph updates to show only the visible curves. The Accent color picker at the bottom recolors the active highlight throughout the tool.
Plot Size, Applied Strip and Info Panel
The S, M, L and XL buttons above the plot set its size, and the readout next to Maximize prints it: S is 500×400, M700×550, L900×850 and XL1100×850. The page opens at 880×460, which is none of the four. The plot never grows wider than its column, so on most screens the buttons mainly change its height. Both curves, the tangent and the markers are redrawn at the new size; the numbers below the plot do not change.
The Applied strip at the bottom of the middle column repeats the current value of x0. Its last tag, x₀ links f and f', is a fixed reminder that one slider position is read on both curves; the numbers themselves are in the at-the-point card above it.
The info panel on the right has two tabs. Explanation is the live one: the family's function and derivative, what x0 controls, the current f(x0) and f′(x0) with the tangent slope they give, and a closing note that links to the section on that family. Concepts is fixed text: what a derivative is, the two ways the tool shows it, and what to look for at extrema and inflection points. For the definition, see what is a derivative.
What Is a Derivative
The derivative of f at the point x0 measures how fast f changes when the input changes a tiny bit. Formally:
f′(x0)=h→0limhf(x0+h)−f(x0)
The fraction inside the limit is the slope of the secant line through the two nearby points (x0,f(x0)) and (x0+h,f(x0+h)). As h shrinks to zero, the secant slope approaches the slope of the tangent line to f at x0 — the derivative.
When the derivative exists at every x in some interval, the collection of values f′(x) forms a new function f′, called the derivative function of f.
For the full theory of derivatives and limit-based definitions, see the derivatives page.
Geometric Meaning - The Tangent Line
The geometric picture: the tangent line to f at x0 is the straight line that just barely touches the curve at one point, sharing the same direction as f there. Its slope is f′(x0), and it passes through (x0,f(x0)). The point-slope form gives:
y=f(x0)+f′(x0)(x−x0)
This is also the best linear approximation to f near x0 — for inputs close to x0, the tangent line predicts f accurately.
Three things to look for as you slide x0 across the graph:
• Steep parts of f — the tangent tilts hard; f′ is far from zero.
• Flat parts of f — the tangent goes horizontal; f′ is at zero.
• Maxima and minima of f — exactly where f′ crosses zero.
For deeper coverage of the tangent line and its applications, see the tangent line page.
From f to f Prime
The derivative isn't just one number at one point — it's a whole new function. As x varies, the slope of the tangent to f at x varies too, tracing out a curve. That curve is the derivative functionf′(x).
The qualitative correspondence between f and f′:
• f increasing ⇔f′>0 (curve of f′ above the x-axis).
• f decreasing ⇔f′<0 (curve of f′ below the x-axis).
• f has a local max or min ⇔f′ crosses zero.
• f concave up ⇔f′ increasing.
• f concave down ⇔f′ decreasing.
• f has an inflection point ⇔f′ has a local max or min.
Sliding x0 through the tool with both curves visible makes this dictionary visible at every step.
Related Concepts
Limits — the foundation derivatives are built on. The derivative is defined as a limit of secant slopes.
Continuity — every differentiable function is continuous, though not the other way around. The absolute value function is continuous at 0 but not differentiable there.
Differentiation rules — power rule, product rule, quotient rule, chain rule. The mechanical toolkit for computing derivatives without going back to the limit definition.
Second derivative — the derivative of f′ itself, written f′′. Measures concavity and acceleration.
Critical points and optimization — finding where f′(x)=0 to locate maxima and minima. The basis of single-variable optimization.
Mean Value Theorem — guarantees that on a smooth interval the average slope equals the instantaneous slope at some point.
Integrals — the inverse operation. Integration recovers f from f′ up to a constant.
Visual tools for calculus — other interactive visualizers covering limits, continuity, Riemann sums, and integrals.