Math With AmarA C A D E M Y
All math animations

High school · Differentiation

From a secant to a tangent

Bring two points together and compare an average rate with the exact derivative.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
From a secant to a tangent. Positive separation h: 1.6. Secant slope: 1.3. Exact tangent slope: 0.5. Slope error h/2: 0.8f(x) = x²/2-2024048xAmber: secant · dashed navy: tangent
The fixed curve is f(x) = x²/2. The graph shows x from −2 to 4 and y from −0.5 to 8; axis scales differ. Only positive h is explored, ending at 0.04. Lines outside the plotting window are clipped.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Positive separation h
1.6
Secant slope
1.3
Exact tangent slope
0.5
Slope error h/2
0.8

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

A secant compares the change in function value between two distinct points. For f(x) = x²/2, its slope between x and x + h is x + h/2. As the positive gap h approaches zero, these slopes approach the tangent slope x. A finite drawing illustrates a limit; it does not reach h = 0.

A relationship to keep

[f(x + h) − f(x)]/h = x + h/2 → f′(x) = x

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Fix the first point

    The teal point stays on f(x) = x²/2 at your chosen x. The amber point is at x + h. Their joining amber line measures average change over that interval.

  2. STEP 2

    Shrink a nonzero interval

    Playback reduces h to 0.04, keeping the denominator nonzero. The displayed secant slope is calculated from the current gap, so it remains slightly different from the exact tangent slope.

  3. STEP 3

    Compare with the limit

    The dashed navy tangent has slope x for every frame. The secant slope error is exactly h/2, which explains both the direction of the difference and why the two lines approach each other.

Your turn to explain

Make a prediction. Test your reasoning.

At x = 1 with h = 0.2, what are the secant slope, tangent slope, and error?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The secant slope is 1.1, the tangent slope is 1, and the difference is 0.1. A small gap gives an approximation, not equality.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Foundations · Calculus

What does a speed reading mean at one instant?

Shrink a time interval to connect average velocity, a difference quotient, and instantaneous velocity.

Grade 12 · Calculus

Derive a power derivative from the difference quotient

Cancel a nonzero increment before taking its limit.