Every graph on this site draws the function and its derivative together, because the pair says more than either curve alone.
Zero crossings mark turning points
Wherever the derivative crosses the axis, the function has a maximum, a minimum or a flat inflection. This is the single most useful correspondence, and it is the one to check first when an answer looks suspicious.
Sign tells direction
Derivative above the axis: the function is rising. Below: falling. A derivative that never changes sign
belongs to a function that never turns — a cubic like x³ − 4x² + 7x − 9 is a good example, because
its parabola derivative misses the axis entirely.
Height tells steepness
The further the derivative is from the axis, the steeper the function. On sin x and cos x this shows
as a quarter-turn offset: the cosine peaks exactly where the sine is climbing fastest.
Blow-ups mark vertical tangents
Where the derivative runs off the top of the frame, the function has a vertical tangent or a pole nearby — the square root at the origin, the arcsine at ±1.