See the derivative as a tangent line slope. Animate the tangent point and plot f(x) alongside f\
Derivative Visualizer plots an entered function together with its derivative, then draws a tangent line at a movable point on the original curve. Animating that point shows how the tangent's slope changes and how each slope becomes a value on the derivative graph. The key distinction is that the vertical coordinate on f'(x) represents slope, not the height of f(x). A high function value can therefore have a zero, negative, or positive derivative.
A concrete input and expected output from the current implementation.
Input
Function: x^2; tangent point: x = 2
Expected output
The function point is (2, 4), the derivative graph is f'(x) = 2x, its value at x = 2 is 4, and the tangent line is y = 4x - 4.
Differentiating x^2 gives 2x, so the slope at x = 2 is 4. A line with slope 4 through (2, 4) satisfies y - 4 = 4(x - 2), which simplifies to y = 4x - 4.
After a function is entered, the display plots y = f(x) and y = f'(x) over the viewing interval. At any horizontal position x = a, the height of the derivative curve represents the slope of the original curve there. Positive derivative values correspond to rising portions of f, negative values to falling portions, and zeros to horizontal tangents when the derivative exists.
The movable point has coordinates (a, f(a)). Its tangent uses slope f'(a), so its line is y = f(a) + f'(a)(x - a). During animation, a changes and the point, line, and corresponding derivative value change together. This connects local steepness on f(x) with a point on the separate derivative curve.
No. The tangent is determined by sharing the function's point and derivative at the selected x-value. It can cross the curve at that point or meet it again elsewhere. Tangency describes local agreement in value and slope, not a requirement that the line stay on one side of the curve or have only one intersection.
No finite derivative exists at a corner when the left-hand and right-hand slopes are different. The picture can help reveal that mismatch, but a line drawn near the corner does not create a derivative there. Similar care is needed at discontinuities, cusps, and vertical tangents, where the usual finite-slope interpretation fails.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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