Premium experiment • derivative lab
MATHEMATICS • CALCULUS
Derivative Visualizer
Animate a forward secant toward the tangent, compare it with a central difference, inspect convergence error and solve for a requested approximation accuracy.
● LIVE LIMIT PROCESSh 0.500 • secant vs tangent
f(x₀)1.000
Exact derivative2.0000
Forward slope2.5000
Central slope2.0000
Forward error5.00e-1
Challenge error4.80e-1
Make the limit visible
The tangent is not presented as magic. A finite secant is compared with the exact derivative while h shrinks, and the error plot records how the approximation improves. Central difference provides a second numerical method for comparison.
Fast investigations
- For x² at x₀ = 1, halve h repeatedly and track forward error.
- Compare forward and central differences for x³.
- Move x₀ on sin(x) and watch the exact slope change sign.
- Use the target-error challenge to estimate how small h must be.
LEARN WITH THE SIMULATION
Understand Derivative Visualizer by experimenting.
START HERE
A derivative measures local rate of change. The visual bridge is the secant line: as a second point approaches the target point, the secant slope approaches the tangent slope.
BUILD THE IDEA
What you should understand first
- Interpret derivative as slope and rate of change.
- See a difference quotient approach a limit.
- Compare analytic derivative values with geometric slopes.
TRY IT YOURSELF
How small must h become before the secant slope closely matches the tangent slope?
- Choose a function and target x-value.
- Start with a visibly large h.
- Reduce h step by step and compare secant and tangent slopes.
Read the visual:
As h approaches zero, the secant slope approaches the derivative when the function is differentiable.
WHY IT WORKS 01
Interpret derivative as slope and rate of change.
A derivative measures local rate of change. The visual bridge is the secant line: as a second point approaches the target point, the secant slope approaches the tangent slope. Focus on the relationship between the controlled variable and the measured response rather than treating the visual change as the explanation.
WHY IT WORKS 02
See a difference quotient approach a limit.
As h approaches zero, the secant slope approaches the derivative when the function is differentiable. Use an extreme or boundary setting after the baseline trial to check whether the same reasoning still holds.
DO NOT MIX THESE UP
Common misconception
A derivative is not simply y/x. Very small numerical h can eventually introduce floating-point effects. Corners and discontinuities can prevent a derivative from existing.
PRACTICE CHALLENGE
How small must h become before the secant slope closely matches the tangent slope?
- Choose a function and target x-value.
- Start with a visibly large h.
- Reduce h step by step and compare secant and tangent slopes.
Reveal the reasoning
As h approaches zero, the secant slope approaches the derivative when the function is differentiable. A derivative is not simply y/x. Very small numerical h can eventually introduce floating-point effects. Corners and discontinuities can prevent a derivative from existing.
MODEL BOUNDARIES
What this simulation assumes
Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.
Model assumptions
- The selected function is evaluated over a finite real domain.
- Slopes are estimated numerically from nearby sampled values or displayed tangents.
- The function is treated as differentiable where a derivative is shown.
Model limitations
- Numerical step size and floating-point precision can affect the displayed slope.
- Derivatives at discontinuities, cusps or other non-differentiable points require interpretation beyond the visualization.
CONNECT IT
Where this matters
- Velocity from position
- Optimization
- Sensitivity analysis
WORDS THAT MATTER
Key vocabulary
- Derivative
- Instantaneous rate of change.
- Secant line
- Line through two points on a curve.
- Tangent line
- Line matching the curve’s local slope at a point.
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