Grade 12 · The product rule
The product rule: A shallow decay with a zero crossing
The product rule: investigate a shallow decay with a zero crossing with linear shift a = -0.5; exponential rate b = -0.25.
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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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 = -0.5; Exponential rate b = -0.25. 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.
- STEP 1
Set up the mathematical model
Name the linear and exponential factors and differentiate each one separately. The starting case is “A shallow decay with a zero crossing.”
- STEP 2
Follow the changing quantity
Move from x=−1 to x=1. Follow the tangent while comparing the two product-rule contributions numerically.
- 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 = -0.5; Exponential rate b = -0.25. Pause the timeline at 100%. Given input x = 1, 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^((-0.25)(1))[1+(-0.25)(1+(-0.5))]=0.681451. Its two terms are 0.778801 and -0.09735; both contributions must be included. Results: Product value: 0.389; Derivative: 0.681; Exponential-factor contribution: -0.097. 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.