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Ex 5.3, 5 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 by Teachoo
Chapter 5 Class 12 Continuity and Differentiability
Concept wise
- Checking continuity at a given point
- Checking continuity at any point
- Checking continuity using LHL and RHL
- Algebra of continous functions
- Continuity of composite functions
- Checking if funciton is differentiable
- Finding derivative of a function by chain rule
-
Finding derivative of Implicit functions
- Finding derivative of Inverse trigonometric functions
- Finding derivative of Exponential & logarithm functions
- Logarithmic Differentiation - Type 1
- Logarithmic Differentiation - Type 2
- Derivatives in parametric form
- Finding second order derivatives - Normal form
- Finding second order derivatives- Implicit form
- Proofs
- Verify Rolles theorem
- Verify Mean Value Theorem
Transcript
Ex 5.3, 5 Find 𝑑𝑦/𝑑𝑥 in, 𝑥2 + 𝑥𝑦 + 𝑦2 = 100 𝑥2 + 𝑥𝑦 + 𝑦2 = 100 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 . 𝑑(𝑥2 + 𝑥𝑦 + 𝑦2)/𝑑𝑥 = (𝑑 (100))/𝑑𝑥 𝑑(𝑥2)/𝑑𝑥 + 𝑑(𝑥𝑦)/𝑑𝑥 + (𝑑(𝑦2))/𝑑𝑥 = (𝑑 (100))/𝑑𝑥 2𝑥^(2−1) + (𝑑(𝑥𝑦))/𝑑𝑥 + (𝑑(𝑦2))/𝑑𝑥 × 𝑑𝑦/𝑑𝑦 = 0 2𝑥 + (𝑑(𝑥𝑦))/𝑑𝑥 + (𝑑(𝑦2))/𝑑𝑦 × 𝑑𝑦/𝑑𝑥 = 0 As (𝑥^𝑛 )^′=𝑛𝑥^(𝑛−1) & derivative of a constant is zero 2𝑥 + (𝑑(𝑥𝑦))/𝑑𝑥 + 2𝑦^(2−1) . 𝑑𝑦/𝑑𝑥 = 0 2𝑥 + (𝑑(𝑥𝑦))/𝑑𝑥 + 2𝑦 . 𝑑𝑦/𝑑𝑥 = 0 Using product rule in 𝑥𝑦 = 𝑥^′ 𝑦+𝑦^′ 𝑥 2𝑥 + ((𝑑(𝑥))/𝑑𝑥 .𝑦+𝑑(𝑦)/𝑑𝑥.𝑥) + 2𝑦 . 𝑑𝑦/𝑑𝑥 = 0 2𝑥 + (1 .𝑦+𝑥.𝑑(𝑦)/𝑑𝑥) + 2𝑦 . 𝑑𝑦/𝑑𝑥 = 0 2𝑥 + 𝑦 + 𝑥 . 𝑑𝑦/𝑑𝑥 + 2𝑦 . 𝑑𝑦/𝑑𝑥 = 0 (2𝑥 + 𝑦) + 𝑑𝑦/𝑑𝑥 . (𝑥 + 2𝑦) = 0 𝑑𝑦/𝑑𝑥 . (𝑥 + 2𝑦) = − (2𝑥 + 𝑦) 𝒅𝒚/𝒅𝒙 = (−( 𝟐𝒙 + 𝒚 ))/( ( 𝒙 + 𝟐𝒚 ))
