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Chain Rule
For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
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Chain Rule
What is it?
For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
In simple words
- For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
- Worked pattern: d(3x+1)5/dx=15(3x+1)4.
- Final check: domain, sign, notation, units, and reasonableness.
Important terms
Chain rule: For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
What you should learn
- State the rule for chain rule.
- Solve a chain rule problem step by step.
- Check the result, restrictions, notation, and units.
Recommended for Class 12
Full school-level explanation
For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
CONCEPT AND RULE
For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
METHOD
For chain rule, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
d(3x+1)5/dx=15(3x+1)4.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve chain rule
How it works
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Remember
- Chain rule
- rule
- worked example
- worked example
How the ideas connect
Worked example: d(3x+1)5/dx=15(3x+1)4.
Deeper understanding
Go beyond memorising Chain Rule. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked example: d(3x+1)5/dx=15(3x+1)4.
Common mistakes
Exam focus
Core Idea
Rule / Formula
Method
Worked Example
Exam focus
Quick revision
- Chain rule
- rule
- worked example
- Chain rule: For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
- CONCEPT AND RULE: For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
- METHOD: For chain rule, identify the given information and required result. Select the matching definition o
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
For long answers on Chain Rule: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- Chain rule
- rule
- worked example
- Chain rule: For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
Important formulas
- Chain rule
- rule
- worked example
Common exam traps
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- A matching-looking formula is not valid until its conditions and domain have been checked.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- Mistake
A matching-looking formula is not valid until its conditions and domain have been checked.
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Chain rule
rule
worked example
Chain rule: For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
- Definition
- For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
- Critical warning
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all d
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Exam Practice
Define
Q1. Define Chain Rule.
Explain
Q2. Explain: CONCEPT AND RULE.
Example
Q3. Give one clear example of Chain Rule and say why it fits.
Mistake
Q4. State one common mistake students make about Chain Rule.
Exam note
Q5. Write a short exam note on Chain Rule: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.