Derivative: Interactive Visualization

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Derivative in this visualization

The derivative of a function $f$ at a point $a$ , written $f'(a)$ or $\frac{df}{dx}$ at $x = a$ , is the slope of the tangent line to the graph there: the instantaneous rate of change of $f$ . It is defined as a limit of slopes of secant lines. This page builds that limit on smooth curves, at corners, cusps, vertical tangents and jumps, and on a position–time graph.

The blue point P sits on the curve at $x = a$ , the red point Q at $x = a + h$ . The yellow line through them is a secant line; its slope is the difference quotient

Difference quotient

\frac{f(a+h) - f(a)}{h} = \frac{\Delta y}{\Delta x}.

The tangent line at P has slope equal to the derivative, when this limit exists and is finite:

Derivative

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.

As Q slides toward P, the secant turns into the tangent and its slope closes in on $f'(a)$ . The limit has to be the same whether Q comes from the left or from the right. The page never lets $h$ reach 0, where the quotient would be $0/0$ . With $x_0$ for $a$ and $\Delta x$ for $h$ , the derivative is also written $f'(x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$ .

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Things to notice

On $x^2$ , $x^3 - 3x$ , $\sin x$ and $e^x$ , the slopes from both sides approach the same number.
$|x|$ at 0: the two sides approach $-1$ and $+1$ . A corner has no derivative, although the function is continuous there.
$x^{2/3}$ at 0: $-\infty$ and $+\infty$ , a cusp.
$\sqrt[3]{x}$ at 0: both sides grow without bound. The tangent is vertical, so there is no finite derivative.
The jump example at 1: a function that is not continuous cannot be differentiable.

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Where it is used

Velocity is the derivative of position, as in the page’s Motion mode. In general the derivative is the rate of change of any quantity: marginal cost in economics, a reaction rate in chemistry, a growth rate in biology.
Near $a$ , the tangent line is the best straight-line approximation, $f(a + h) \approx f(a) + f'(a)\,h$ . Newton’s method for solving equations follows tangent lines like this one.
At a maximum or minimum inside an interval, a differentiable function has $f' = 0$ : the way calculus finds the best value.

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Controls

Drag Q, or use the |h| fader or h → 0 button in the panel, to change $h$ . Left / Right sets its sign.
Drag P or turn the Point knob to move $a$ . Drag the background to pan; use + and − to zoom.
Compare & record opens a window plotting the secant slope against $h$ on both sides and collecting the values left by the approach. Values are still collected while the window is closed.
Motion turns the curve into a position $s(t)$ : the secant slope becomes an average velocity, the derivative the instantaneous velocity $v(t) = s'(t)$ .
Presentation mode opens a full-screen view for projectors, with large type and keyboard control (Space runs h → 0, Page Up / Page Down step through the highlights, H hides the derivative).

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Derivative in this visualization

Things to notice

Where it is used

Controls

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