Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Differentiate an inverse composition at matching input-output points.
- Justify the conclusion "g′(8)=1/f′(2)=1/12" using the stated assumptions.
Before you start
Chain rule and inverse functions.
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
If f(x)=x³ and g=f⁻¹, find g′(8).
Why this math matters
Differentiate an inverse composition at matching input-output points. 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:
- The derivative of f at the matching point is nonzero.
- A differentiable inverse exists locally near that output.
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.
Relate the sensitivity of a function to that of its inverse
PausedQuestion: Start with the question. Paused.
Question
Start with the question
If f(x)=x³ and g=f⁻¹, find g′(8).
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
f(2)=8, so g(8)=2
The inverse derivative must be evaluated at the matching original point.
Work through the mathematics
f′(g(y))g′(y)=1
Differentiating f(g(y))=y gives the reciprocal-slope relation.
Check the conclusion
g′(8)=1/f′(2)=1/12
The inverse changes slowly where the forward map has a large nonzero derivative.
The result
g′(8)=1/f′(2)=1/12
The inverse changes slowly where the forward map has a large nonzero derivative.
Common mistakes to catch
- Use f′ at g(y), not at y.
- A one-to-one function can still have an inverse that is not differentiable at some points.
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
Can the same reciprocal formula give a finite derivative at y=0?
Show a hint
f′(0)=0.
Reveal answer and explanation
No
The cube-root graph has an unbounded slope at zero.
Practice 2
For f(x)=5x+1, what is the inverse derivative?
Show a hint
Invert its constant slope.
Reveal answer and explanation
1/5
Reversing the linear scale reverses the sensitivity factor.
Take the idea with you
Compare measurement sensitivity before and after inverting a calibration curve.
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 when the base and exponent both vary
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