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Grade 12 · The product rule

The product rule: Cancellation at the origin

The product rule: investigate cancellation at the origin with linear shift a = 1; exponential rate b = -1.

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

Watch the relationship

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The product rule: Cancellation at the origin. Input x: -1. Product value: 0. Derivative: 2.718. Exponential-factor contribution: 0Two contributions to one derivative01-101yx → · labeled axes rescale to this model
The exponential is always positive and both factors are differentiable on the real line. The domain shown is finite. The b=0 case is a constant exponential factor and reduces to a shifted line.

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.

Input x
-1
Product value
0
Derivative
2.718
Exponential-factor contribution
0

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Understand what you are seeing

The idea behind the motion.

Both factors in a product can change at once. The derivative must account for changing the linear factor and changing the exponential factor. These contributions may reinforce each other or cancel, so multiplying the two individual derivatives does not produce the derivative of their product. This investigation starts with Linear shift a = 1; Exponential rate b = -1. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

f=(x+a)e^(bx); f′=e^(bx)+b(x+a)e^(bx)

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

  1. STEP 1

    Set up the mathematical model

    Name the linear and exponential factors and differentiate each one separately. The starting case is “Cancellation at the origin.”

  2. STEP 2

    Follow the changing quantity

    Move from x=−1 to x=1. Follow the tangent while comparing the two product-rule contributions numerically.

  3. STEP 3

    Explain and test the result

    Choose parameters that make one contribution vanish or cancel the other. Check that a zero derivative need not mean either original factor is zero.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Linear shift a = 1; Exponential rate b = -1. Pause the timeline at 20%. Given input x = -0.6, calculate product value, derivative, exponential-factor contribution. Show the substitution into the displayed formula.

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

Compare your explanation

The product rule gives e^((-1)(-0.6))[1+(-1)(-0.6+(1))]=1.093271. Its two terms are 1.822119 and -0.728848; both contributions must be included. Results: Product value: 0.729; Derivative: 1.093; Exponential-factor contribution: -0.729. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 12 · Calculus

Differentiate a product with two changing factors

Include one contribution for the change in each factor.