Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Include one contribution for the change in each factor.
- Justify the conclusion "At x=π, f′(π)=−π²" using the stated assumptions.
Before you start
Basic derivatives and products.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Differentiate f(x)=x²sin x.
Why this math matters
Include one contribution for the change in each factor. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Angles in sin x are radians.
- Both factors are differentiable.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Differentiate a product with two changing factors
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Differentiate f(x)=x²sin x.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
One factor is x² with derivative 2x; the other is sin x with derivative cos x
Both factors vary with the same input.
Work through the mathematics
f′=(2x)sin x+x²cos x
The product rule adds the two first-order contributions.
Check the conclusion
At x=π, f′(π)=−π²
The sine term vanishes and cosine contributes minus one.
The result
At x=π, f′(π)=−π²
The sine term vanishes and cosine contributes minus one.
Common mistakes to catch
- The product rule is a sum, not a product of derivatives.
- A factor that vanishes at one point may still contribute through its derivative.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Differentiate x exp(x).
Show a hint
Apply the same two-term rule.
Reveal answer and explanation
exp(x)+x exp(x)
Each factor contributes once.
Practice 2
Is the derivative the product of the two derivatives?
Show a hint
Test x·x.
Reveal answer and explanation
No
That would give one, but the derivative of x² is 2x.
Take the idea with you
Separate changing size from changing oscillation when analyzing a modulated signal.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Differentiate a normalized ratio
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