Finding second order derivatives- Implicit form

Chapter 5 Class 12 Continuity and Differentiability
Concept wise

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### Transcript

Ex 5.7, 14 If π¦= γAπγ^ππ₯ + γBπγ^ππ₯, show that π2π¦/ππ₯2 β (π+π) ππ¦/ππ₯ + πππ¦ = 0 π¦= γAπγ^ππ₯ + γBπγ^ππ₯ Differentiating π€.π.π‘.π₯ ππ¦/ππ₯ = (π(γAπγ^ππ₯ " + " γBπγ^ππ₯))/ππ₯ ππ¦/ππ₯ = (π(γAπγ^ππ₯))/ππ₯ + (π(γBπγ^ππ₯))/ππ₯ ππ¦/ππ₯ = A . π^ππ₯. (π(ππ₯))/ππ₯ + B . π^ππ₯ (π(ππ₯))/ππ₯ ππ¦/ππ₯ = A . π^ππ₯. π + B . π^ππ₯. π ππ¦/ππ₯ = π΄ππ^ππ₯ + π΅ππ^ππ₯ Again Differentiating π€.π.π‘.π₯ π/ππ₯ (ππ¦/ππ₯) = π(π΄ππ^ππ₯ " + " π΅ππ^ππ₯ " " )" " /ππ₯ (π^2 π¦)/(ππ₯^2 ) = π(π΄ππ^ππ₯ )/ππ₯ + π(π΅ππ^ππ₯ )" " /ππ₯ (π^2 π¦)/(ππ₯^2 ) = π΄π π(π^ππ₯ )/ππ₯ + π΅π π(π^ππ₯ )" " /ππ₯ (π^2 π¦)/(ππ₯^2 ) = π΄π . π^(ππ₯ ). π(ππ₯ )/ππ₯ + π΅π . π^ππ₯ . π(ππ₯)/ππ₯ (π^2 π¦)/(ππ₯^2 ) = π΄ππ^(ππ₯ ) . π+π΅ππ^ππ₯ . π (π^2 π¦)/(ππ₯^2 ) = π΄π2π^(ππ₯ )+π΅π2π^ππ₯ We need to prove (π^2 π¦)/(ππ₯^2 ) β (π+π) ππ¦/ππ₯ + πππ¦ = 0 Solving LHS (π^2 π¦)/(ππ₯^2 ) β (π+π) ππ¦/ππ₯ + πππ¦ = (π΄π2π^(ππ₯ )+π΅π2π^ππ₯) β (π+π) (π΄ππ^(ππ₯ )+π΅ππ^ππ₯) + ππ (π΄π^(ππ₯ )+π΅π^ππ₯) = π΄π2π^(ππ₯ )+π΅π2π^ππ₯ β π(π΄ππ^(ππ₯ )+π΅ππ^ππ₯) β π(π΄ππ^(ππ₯ )+π΅ππ^ππ₯) + ππ π΄π^(ππ₯ )+πππ΅π^ππ₯ = π΄π2π^(ππ₯ )+π΅π2π^ππ₯ βπ΄π2π^(ππ₯ )β π΅πππ^ππ₯ β π΄πππ^(ππ₯ ) + π΅π2π^ππ₯+ ππ π΄π^(ππ₯ )+πππ΅π^ππ₯ = π΄π2π^(ππ₯ )β π΄π2π^(ππ₯ ) + π΅π2π^ππ₯ βπ΅π2π^ππ₯ β π΅πππ^ππ₯ + π΅πππ^ππ₯ β π΄πππ^(ππ₯ ) + π΄πππ^(ππ₯ ) = 0 = RHS Hence proved

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.