Aligned function and derivative panels with shared x-cursorUpdates with every input
Math & Statistics
Derivative Graphing Calculator
Graph a quartic polynomial together with its derivative on aligned axes, locate real stationary points numerically, classify local turning behavior, and inspect the selected x-coordinate.
Selected f(x) -
Selected f'(x) -
Selected f''(x) -
Direction at selected x -
Stationary points in window -
Local minima -
Local maxima -
Displayed x-window -
LINKED FUNCTION VIEW
Aligned function and derivative panels with shared x-cursor
Zeros of the derivative line up vertically with horizontal tangents and classified turning points on the function.
STATIONARY-POINT REGISTER
Numerically isolated derivative roots and local classification
The table evaluates f, f prime, and f double-prime at each detected root in the visible interval.
| Point | x | f(x) | f'(x) | f''(x) | Classification |
|---|
GRAPH WINDOW
Choose coefficients and an interval that contains the behavior of interest
- Enter zero for omitted polynomial terms.
- Set xmin below xmax.
- Choose a window narrow enough to see turning behavior.
- Move the selected x cursor for live slope.
- Use the table rather than estimating root coordinates from pixels.
VERTICAL ALIGNMENT
A derivative zero corresponds to a horizontal tangent, not automatically a turning point
A sign change in f' identifies a local direction reversal. A derivative can also touch zero without changing sign, producing a stationary inflection or flatter contact.
The panels share the same x-axis so each derivative feature can be traced vertically to the original function. Separate y-scales prevent one curve from flattening the other.
Subject fundamentals
Five ideas that control this calculation
The top curve is a quartic
Five coefficients define f(x)=ax to the fourth+bx cubed+cx squared+dx+e over the entered viewing interval.
The lower curve is analytic
The derivative is calculated from the coefficients, not estimated from plotted pixels or neighboring samples.
Roots are isolated numerically
The interval is scanned for derivative sign changes and each bracket is refined, so only stationary points inside the chosen window are reported.
Classification uses curvature
Positive f'' at a derivative root indicates a local minimum; negative f'' indicates a local maximum; near-zero curvature requires extra inspection.
Shared x-position enables comparison
The selected marker and every stationary point align vertically across both panels even though their y-scales are intentionally separate.
GRAPH-TO-DERIVATIVE LINK
Differentiate symbolically, then isolate derivative zeros in the selected window
The quartic derivative is cubic. Sign changes and near-zero samples are refined by bisection; the second derivative supports local classification.
Detailed calculation process and general formulas
f(x)=ax^4+bx^3+cx^2+dx+ef'(x)=4ax^3+3bx^2+2cx+df''(x)=12ax^2+6bx+2cstationary point: f'(x*)=0Symbols, meanings, and units
- a,b,c,d,e
- quartic coefficientscoefficient-dependent
- x*
- stationary-point x-coordinatex units
- f'(x)
- slope functiony units per x unit
- f''(x*)
- curvature classifier at a stationary pointy units per x unit squared
- xmin,xmax
- visible search and graph intervalx units
GRAPH DIAGNOSTICS
Read three signals together
No single plotted cue gives the full local classification.
Function shape
-The upper panel shows height and turning geometry.
Derivative sign
-The lower panel shows increasing and decreasing intervals.
Curvature
-The second derivative helps classify isolated stationary points.
Decision takeaway: Confirm a stationary point with numerical values; do not rely on line thickness at the axis.
GRAPHING LIMITS
Cases that require extra care
- Repeated derivative roots
- Very narrow features
- Large coefficient scaling
- Roots outside the window
- Near-flat intervals
- Floating-point resolution
Applied decisions
Why linked panels outperform a single curve
Multiple turning points
A quartic rises, falls, and rises again in the selected interval.
What the result clarifies: Derivative sign bands identify each monotonic section.
Stationary shoulder
The derivative approaches zero without a clear sign reversal.
What the result clarifies: The table and second derivative prevent an automatic maximum/minimum label.
Worked default scenario
Current-input substitution and reconciliation
Key terms
Glossary for interpreting the result
- Quartic
- A polynomial with highest nonzero power four.
- Derivative graph
- The curve y=f'(x), showing instantaneous slope across x.
- Stationary point
- An x-value where f'(x)=0.
- Bisection
- A root-refinement method that repeatedly halves a sign-change interval.
- Local extremum
- A nearby maximum or minimum identified from derivative and curvature behavior.
- Viewing window
- The entered xmin-to-xmax interval included in the plot and root search.
Method references
References for this calculator's specific method
Scope and limitations
Roots are numerically searched only inside the entered window. Closely repeated roots or extreme scaling may require a dedicated algebra system or higher-precision numerical method.
Derivative Graphing Calculator | Function, Slope Field, and Stationary Points FAQ
Why use separate y-scales?
Function values and slopes can have very different magnitudes; aligned x-axes preserve the relationship without flattening either curve.
Is every f prime zero a maximum or minimum?
No. Classification also depends on sign change or higher-order behavior.
Can a stationary point lie outside the graph?
Yes. Only the entered interval is searched and displayed.
Why may a root be reported as approximate?
The derivative root is isolated numerically using finite floating-point steps.