Derivative Graph Calculator

A derivative is a function in its own right, and it has a graph — one that tells the story of the original function's behaviour at a glance. Where the derivative's graph sits above the axis, the original climbs; where it dips below, the original falls; where it crosses, the original levels off. Plotting the two together by hand is slow and error-prone, especially when you need the picture to reason about turning points. This calculator draws the function and its derivative on the same coordinate plane so you can read that story directly.

Quick Answer

The derivative graph is the plot of f′(x) against x, and it encodes the original graph's behaviour point by point: f′ positive → f increasing; f′ negative → f decreasing; f′ = 0 → f has a horizontal tangent (a candidate turning point).

Reading Two Graphs as One Story

The key insight is that the derivative graph is not a picture of heights but a picture of slopes. Pick any x-value: the height of the derivative graph there equals the slope of the tangent to the original curve at that same x. A steep climb on the original appears as a tall peak on the derivative; a gentle fall appears as a shallow dip below the axis. The two graphs therefore move in a linked rhythm — when the original's slope is at its steepest, the derivative graph is at its highest or lowest point, which is where the original has an inflection point.

Some correspondences are worth memorising because they let you sketch either graph from the other in an exam. A local maximum or minimum of f sits directly above a zero crossing of f′, and the crossing direction tells you which: f′ going from positive to negative signals a maximum, negative to positive a minimum. A corner or cusp on f — where the slope jumps — shows up as a gap or jump in f′. And the degree drops by one: a cubic's derivative is a parabola, a parabola's derivative is a line, which gives you a fast shape check before any detailed reading.

How to Use the Derivative Graph Calculator

1. Enter the function f(x) exactly as written, with brackets around grouped terms.
2. Choose a viewing window wide enough to include the features you care about — turning points and intercepts are easily missed in a narrow window.
3. Run the calculation and identify which plotted curve is the function and which is its derivative before interpreting anything.
4. Find where the derivative graph crosses the x-axis; drop vertically to the original graph to locate its turning points.
5. Check a specific slope by picking an x-value and comparing the derivative graph's height there with the steepness of the original curve.
6. If you need the derivative's formula as well as its picture, note it from the result and verify a turning point numerically.

A Worked Example: f(x) = x³ − 3x and Its Parabola of Slopes

The derivative is f′(x) = 3x² − 3 = 3(x − 1)(x + 1), a parabola opening upwards with zeros at x = −1 and x = 1. Now read the pair of graphs. The derivative is positive outside the interval [−1, 1], so the cubic climbs steeply on the left, and positive again after x = 1, so the cubic climbs again on the right. Between −1 and 1 the derivative is negative, so the cubic falls through the middle. At x = −1 the derivative crosses from positive to negative, marking the local maximum f(−1) = (−1) + 3 = 2. At x = 1 it crosses back, marking the local minimum f(1) = 1 − 3 = −2. At x = 0 the derivative reaches its lowest value, −3, which is the cubic's steepest descent and its inflection point. Every feature of one graph is explained by the other — that is the check to perform on any pair you produce.

Misreadings That Flip the Whole Interpretation

The most damaging mistake is reading the derivative graph as if it were the function: seeing the derivative's maximum and reporting it as the function's maximum, when it actually marks the function's steepest slope. A second is assuming every zero of f′ is a turning point — a derivative can touch zero without changing sign (as f′(x) = 3x² does at the origin), leaving an inflection rather than an extremum. Window choice causes quieter errors: zoomed too far out, a shallow local feature disappears; too far in, a parabola masquerades as a line. Always cross-check at least one feature algebraically. The Extreme Points Calculator pins down the exact coordinates the graph suggests, and the Second Derivative Calculator confirms whether a stationary point is a maximum, a minimum, or neither.

A Correspondence Table Worth Internalising

Most graph-reading questions reduce to a handful of correspondences, and it helps to state them in one place. Work through them against the paired graphs until each feels obvious:

• f is increasing on an interval exactly where the f′ graph lies above the x-axis there.
• f is decreasing exactly where the f′ graph lies below the x-axis.
• A peak or trough of f aligns vertically with an x-intercept of f′ where f′ changes sign.
• The steepest points of f — its inflection points — align with the peaks and troughs of f′.
• Where f′ is increasing, f is concave up; where f′ is decreasing, f is concave down.
• A corner on f appears as a jump or gap in f′; a vertical tangent on f sends f′ shooting off to infinity.

Notice that every row converts a question about f into a question about the graph of f′ that can be answered by looking rather than computing. Exam questions that hand you the derivative's graph and ask about the function — a common format — are testing precisely this table, and the calculator's paired display is the fastest way to drill it: predict each row's effect before you look, then look.

Graphs First, Algebra to Confirm

The strongest workflow in curve sketching is picture, then proof: let the paired graphs show you where the action is, then justify each feature with derivatives and tests. Use this calculator to build that intuition, and the plain Derivative Calculator whenever you need the symbolic derivative without the picture.