MathematicsChapter 2: Limit, Continuity and Derivative
Derivative From First Principles
f'(x)=lim(h->0)[f(x+h)-f(x)]/h measures instantaneous rate of change.
Learning objectives
- State the rule for derivative from first principles.
- Solve a derivative from first principles problem step by step.
- Check the result, restrictions, notation, and units.
Definition
f'(x)=lim(h->0)[f(x+h)-f(x)]/h measures instantaneous rate of change.
Where / symbols
- Derivative from first principles
- f'(x)=lim(h->0)[f(x+h)-f(x)]/h measures instantaneous rate of change.
CONCEPT AND RULE
f'(x)=lim(h->0)[f(x+h)-f(x)]/h measures instantaneous rate of change.
METHOD
For derivative from first principles, 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
For f(x)=x2, expansion gives f'(x)=2x.
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 derivative from first principles
- 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:
- For f(x)=x2, expansion gives f'(x)=2x.
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
- Derivative from first principles
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Derivative From First Principles
Sticky Notes · Formula and Worked Revision
Core Idea
- f'(x)=lim(h->0)[f(x+h)-f(x)]/h measures instantaneous rate of change.
Rule / Formula
- f'(x)=lim(h->0)[f(x+h)-f(x)]/h measures instantaneous rate of change.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- For f(x)=x2, expansion gives f'(x)=2x.
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
- f'(x)=lim(h->0)[f(x+h)-f(x)]/h measures instantaneous rate of change.
- Worked pattern: For f(x)=x^2, expansion gives f'(x)=2x.
- Final check: domain, sign, notation, units, and reasonableness.
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