Derivative as a Tangent Slope

Sweep x and watch the tangent line’s slope get drawn out as the derivative curve f′(x) below.

This interactive derivative visualization shows what a derivative really is: the slope of the tangent line to a curve at a point. As you sweep the input $x$, a tangent line rides along the function $f(x)$, and its slope is plotted out beneath it to trace the derivative curve $f'(x)$ — so you literally watch one curve generate another.

The derivative is defined as the limit of the slope of a secant line as two points on the curve slide together: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Where the function rises steeply the tangent tilts up and the derivative is large and positive; where the function is flat — at a peak or a valley — the tangent is horizontal and the derivative crosses zero. Watching those zero-crossings line up with the maxima and minima of the original curve is the moment the concept clicks.

Switch between functions to see how the shape of $f(x)$ determines the shape of $f'(x)$, and step through the sweep slowly to connect each tangent slope to a single point on the derivative curve. It's a visual, hands-on companion to introductory calculus that makes differentiation something you can see happen rather than just a rule to memorize.