MathematicsChapter 2: Limit, Continuity and Derivative
Chain Rule
For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
Learning objectives
- State the rule for chain rule.
- Solve a chain rule problem step by step.
- Check the result, restrictions, notation, and units.
Definition
For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
Mathematical form
y=f(g(x))
dx=15(3x+1)4.
Where / symbols
- 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 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
- 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.
Worked example
- Worked example:
- d(3x+1)5/dx=15(3x+1)4.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Chain rule
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Chain Rule
Sticky Notes · Formula and Worked Revision
Core Idea
- For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
Rule / Formula
- For y=f(g(x)), y'=f'(g(x))g'(x); differentiate outside, keep inside, multiply by inside derivative.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- d(3x+1)5/dx=15(3x+1)4.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- 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.
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