Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Use a logarithm to turn a variable power into a product.
- Justify the conclusion "y′=xˣ(ln x+1)" using the stated assumptions.
Before you start
Logarithms, product rule, and chain rule.
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 y=xˣ for x>0.
Why this math matters
Use a logarithm to turn a variable power into a product. 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 positive.
- xˣ is interpreted through exp(x ln x).
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 when the base and exponent both vary
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Differentiate y=xˣ for x>0.
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
ln y=x ln x
Positivity permits taking a real logarithm.
Work through the mathematics
y′/y=ln x+1
Differentiate the left by the chain rule and the right by the product rule.
Check the conclusion
y′=xˣ(ln x+1)
Multiply by the original y to express the derivative in x.
The result
y′=xˣ(ln x+1)
Multiply by the original y to express the derivative in x.
Common mistakes to catch
- The formula is not obtained by treating the exponent as fixed.
- The stated real domain matters for taking logarithms.
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
Find y′ at x=1.
Show a hint
ln1=0 and 1¹=1.
Reveal answer and explanation
One
The product becomes 1·(0+1).
Practice 2
Why is the usual constant-exponent power rule insufficient?
Show a hint
The exponent also changes.
Reveal answer and explanation
It omits a contribution from the changing exponent
Logarithmic differentiation handles both sources of variation.
Take the idea with you
Identify which differentiation rule fits a variable appearing in both a base and its exponent.
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.
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