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Ex 5.4, 2 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 by Teachoo
Finding derivative of Exponential & logarithm functions
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
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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 5.4, 2 (Method 1) Differentiate 𝑤.𝑟.𝑡. x in , 𝑒^(sin^(−1) 𝑥)Let 𝑦 = 𝑒^(sin^(−1) 𝑥) Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑(𝑦)/𝑑𝑥 = 𝑑(𝑒^(sin^(−1) 𝑥) )/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑒^(sin^(−1) 𝑥) . 𝑑(sin^(−1) 𝑥)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑒^(sin^(−1) 𝑥) . (1/√(1 − 𝑥^2 )) 𝒅(𝒚)/𝒅𝒙 = 𝒆^(〖𝒔𝒊𝒏〗^(−𝟏) 𝒙)/√(𝟏−𝒙^𝟐 ) (𝑑(𝑒^𝑥 )/𝑑𝑥 " = " 𝑒^𝑥 " " ) Ex 5.4, 2 (Method 2) Differentiate 𝑤.𝑟.𝑡. x in , 𝑒^(sin^(−1) 𝑥)Let 𝑦 = 𝑒^(sin^(−1) 𝑥) Let sin^(−1) 𝑥=𝑡 𝑦 = 𝑒^𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑(𝑦)/𝑑𝑥 = 𝑑(𝑒^𝑡 )/𝑑𝑥 We need 𝑑𝑡 in denominator, so multiplying & Dividing by 𝑑𝑡 . 𝑑𝑦/𝑑𝑥= 𝑑(𝑒^𝑡 )/𝑑𝑥 × 𝑑𝑡/𝑑𝑡 𝑑𝑦/𝑑𝑥= 𝑑(𝑒^𝑡 )/𝑑𝑥 × 𝑑𝑡/𝑑𝑡 𝑑𝑦/𝑑𝑥= 𝑑(𝑒^𝑡 )/𝑑𝑡 × 𝑑𝑡/𝑑𝑥 𝑑𝑦/𝑑𝑥= 𝑒^𝑡 × 𝑑𝑡/𝑑𝑥 Putting value of 𝑡 𝑑𝑦/𝑑𝑥= 𝑒^(sin^(−1) 𝑥) × 𝑑(sin^(−1) 𝑥)/𝑑𝑥 𝑑𝑦/𝑑𝑥= 𝑒^(sin^(−1) 𝑥) × 1/√(1 − 𝑥^2 ) 𝒅𝒚/𝒅𝒙 = 𝒆^(〖𝒔𝒊𝒏〗^(−𝟏) 𝒙)/√(𝟏 − 𝒙^𝟐 )
