Derivative — the instantaneous rate of change of f at a point, written f′(x) or dxdf. Equal to the slope of the tangent line to f at that point.
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 derivative f′ 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
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.
The x₀ Slider
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
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
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.
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.