📚 Key Concepts
Continuity- Left hand limit = Right hand limit = Function value at point
- f(x) is continuous at x=a if:
- f(a) exists
- lim x→a f(x) exists
- lim x→a f(x) = f(a)
Differentiability- f'(x) = lim h→0 [f(x+h) - f(x)]/h
- If f(x) is differentiable at a point, it must be continuous there
- Reverse isn't true (e.g., |x| at x=0)
Special Functions- Exponential: (eˣ)' = eˣ
- Logarithmic: (ln x)' = 1/x
- Trigonometric derivatives:
- (sin x)' = cos x
- (cos x)' = -sin x
- (tan x)' = sec² x
- (cot x)' = -cosec² x
- (sec x)' = sec x tan x
- (cosec x)' = -cosec x cot x
🎯 Key Formulas
Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x)Product Rule: (uv)' = u'v + uv'Quotient Rule: (u/v)' = (u'v - uv')/v²Parametric:- dy/dx = (dy/dt)/(dx/dt)
- d²y/dx² = (d²y/dt² · dx/dt - dy/dt · d²x/dt²)/(dx/dt)³
⚠️ Common Mistakes to Avoid
Forgetting chain ruleWrong application of product/quotient rulesIncorrect continuity testingMixing up derivative rulesWrong parametric differentiationNot checking all continuity conditions
📖 Knowledge Prerequisites
LimitsBasic functionsAlgebraic operationsUnderstanding of graphs
💡 Tips for Students
Always check continuity before differentiabilityPractice chain rule with complex functionsDraw graphs to understand continuityLearn standard derivatives thoroughlyUse systematic approach for complicated functions
👉 Practice Recommedations
Test continuity at pointsFind derivatives using different rulesSolve parametric differentiation problemsPractice second order derivativesApply in practical problemsVerify differentiability at points
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