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Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Cancel a nonzero increment before taking its limit.
- Justify the conclusion "As h→0, f′(x)=3x²" using the stated assumptions.
Before you start
Polynomial expansion and limits.
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
Derive the derivative of f(x)=x³ from first principles.
Why this math matters
Cancel a nonzero increment before taking its limit. 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 fixed while h approaches zero.
- The polynomial is defined for all real inputs.
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.
Derive a power derivative from the difference quotient
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Derive the derivative of f(x)=x³ from first principles.
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
[(x+h)³−x³]/h=(3x²h+3xh²+h³)/h
Expand the shifted cube before cancelling.
Work through the mathematics
For h≠0, the quotient equals 3x²+3xh+h²
Cancellation removes the indeterminate zero-over-zero form.
Check the conclusion
As h→0, f′(x)=3x²
The remaining h terms vanish, giving a formula for every real x.
The result
As h→0, f′(x)=3x²
The remaining h terms vanish, giving a formula for every real x.
Common mistakes to catch
- A secant quotient with nonzero h is not yet the derivative.
- Expand every mixed term of the cube.
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
What is the tangent slope at x=−2?
Show a hint
Substitute into 3x².
Reveal answer and explanation
Twelve
Squaring the negative input makes the slope positive.
Practice 2
Why not set h=0 at the start?
Show a hint
The original quotient divides by h.
Reveal answer and explanation
It would be undefined
The derivative uses a limit through nonzero increments.
Take the idea with you
Compare a symbolic derivative with slopes from shrinking input intervals.
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 product with two changing factors
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