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Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Apply the quotient rule and inspect its retained domain.
- Justify the conclusion "f′=(1−x²)/(1+x²)²" using the stated assumptions.
Before you start
Product rule and powers.
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)=x/(1+x²).
Why this math matters
Apply the quotient rule and inspect its retained domain. 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:
- x is real.
- The denominator 1+x² never vanishes.
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 normalized ratio
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Differentiate f(x)=x/(1+x²).
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
Numerator derivative is one; denominator derivative is 2x
Both parts of the ratio vary.
Work through the mathematics
f′=[(1+x²)−x(2x)]/(1+x²)²
The numerator order is denominator times numerator derivative minus numerator times denominator derivative.
Check the conclusion
f′=(1−x²)/(1+x²)²
The denominator is always positive, so the derivative's sign depends only on 1−x².
The result
f′=(1−x²)/(1+x²)²
The denominator is always positive, so the derivative's sign depends only on 1−x².
Common mistakes to catch
- Reversing numerator order reverses the derivative sign.
- The denominator is squared after applying the rule.
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
Where is f increasing?
Show a hint
Solve 1−x²>0.
Reveal answer and explanation
On (−1,1)
The squared denominator does not change the sign.
Practice 2
Where are the stationary points?
Show a hint
Set the derivative numerator to zero.
Reveal answer and explanation
x=−1 and x=1
The original function and derivative are both defined there.
Take the idea with you
Analyze a response that rises initially but decreases after its normalization term dominates.
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 relation without solving it into one global function
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