Grade 12 · The product rule · 361 of 600
The product rule · Linear shift a=2; Exponential rate b=-0.25
Evaluate at x=0.2. Find the product value, its derivative, and the contribution from differentiating the exponential factor. Givens: Linear shift a=2; Exponential rate b=-0.25.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Let f(x)=(x+(2))e^((-0.25)x). Evaluate at x=0.2. Find the product value, its derivative, and the contribution from differentiating the exponential factor.
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A starting point
Differentiate one factor at a time: f′=e^(bx)+b(x+a)e^(bx).
Work through the reasoning
Step 1
Identify the model and target
Let f(x)=(x+(2))e^((-0.25)x). Evaluate at x=0.2. The governing relation is f=(x+a)e^(bx); f′=e^(bx)+b(x+a)e^(bx). Differentiate one factor at a time: f′=e^(bx)+b(x+a)e^(bx).
Step 2
Substitute and calculate
The product rule gives e^((-0.25)(0.2))[1+(-0.25)(0.2+(2))]=0.428053. Its two terms are 0.951229 and -0.523176; both contributions must be included.
Step 3
Check the mathematical meaning
The two derivative contributions are 0.951229 and -0.523176. Their sum is 0.428053, including any cancellation when the second contribution is negative. Input x: 0.2; Product value: 2.093; Derivative: 0.428; Exponential-factor contribution: -0.523. Decimal values are rounded, so use unrounded intermediate values.
The answer
The product rule gives e^((-0.25)(0.2))[1+(-0.25)(0.2+(2))]=0.428053. Its two terms are 0.951229 and -0.523176; both contributions must be included. The two derivative contributions are 0.951229 and -0.523176. Their sum is 0.428053, including any cancellation when the second contribution is negative. Animation check: Input x: 0.2; Product value: 2.093; Derivative: 0.428; Exponential-factor contribution: -0.523. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
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. Let f(x)=(x+(2))e^((-0.25)x). Evaluate at x=0.2. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
f=(x+a)e^(bx); f′=e^(bx)+b(x+a)e^(bx)
03 · Reflect and transfer
Explain what changes and why.
If the exponential rate becomes zero, what do the function and its derivative reduce to?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.