Ex 5.7, 15 - If y = 500 e7x + 600 e-7x, show d2y/dx2 = 49y - Finding second order derivatives- Implicit form

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.7, 15 If 𝑦= γ€–500𝑒〗^7π‘₯+ γ€–600𝑒〗^(βˆ’7π‘₯), show that 𝑑2𝑦/𝑑π‘₯2 = 49𝑦 𝑦= γ€–500𝑒〗^7π‘₯+ γ€–600𝑒〗^(βˆ’7π‘₯) Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑𝑦/𝑑π‘₯ = (𝑑(γ€–500𝑒〗^7π‘₯ "+" γ€–600𝑒〗^(βˆ’7π‘₯) " " ))/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = (𝑑 (γ€–500𝑒〗^7π‘₯))/𝑑π‘₯ + (𝑑 (γ€–600𝑒〗^(βˆ’7π‘₯)))/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 500 (𝑑 (𝑒^7π‘₯))/𝑑π‘₯ + 600 (𝑑 (𝑒^(βˆ’7π‘₯)))/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 500 . 𝑒^7π‘₯. (𝑑 (7π‘₯))/𝑑π‘₯ + 600 . 𝑒^(βˆ’7π‘₯) . (𝑑 (βˆ’7π‘₯))/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 500 . 𝑒^7π‘₯. 7 + 600 . 𝑒^(βˆ’7π‘₯) . (βˆ’7) 𝑑𝑦/𝑑π‘₯ = 500 . 7 . 𝑒^7π‘₯ βˆ’ 600 . 7 . 𝑒^(βˆ’7π‘₯) Again Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑/𝑑π‘₯ (𝑑𝑦/𝑑π‘₯) = 𝑑(500 . 7 . 𝑒^7π‘₯ " βˆ’" γ€– 600 . 7 . 𝑒〗^(βˆ’7π‘₯) )" " /𝑑π‘₯ (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = 𝑑(500 Γ— 7 . 𝑒^7π‘₯ )/𝑑π‘₯ + 𝑑(γ€–600 Γ—7𝑒〗^(βˆ’7π‘₯) )" " /𝑑π‘₯ (𝑑^2 𝑦)/(𝑑π‘₯^2 ) = 500 Γ— 7 𝑑(𝑒^7π‘₯ )/𝑑π‘₯ βˆ’ 600 Γ— 7 𝑑(𝑒^(βˆ’7π‘₯) )/𝑑π‘₯ = 500 Γ— 7 Γ— 𝑒^7π‘₯ 𝑑(7π‘₯)/𝑑π‘₯ βˆ’ 600 Γ— 7 Γ— 𝑒^(βˆ’7π‘₯) 𝑑(βˆ’7π‘₯)/𝑑π‘₯ = 500 Γ— 7 Γ— 𝑒^7π‘₯. 7 βˆ’ 600 Γ— 7 Γ— 𝑒^(βˆ’7π‘₯) (βˆ’7) = 500 Γ— 7 Γ— 7𝑒^7π‘₯ βˆ’ 600 Γ— 7 Γ— 7 Γ— 𝑒^(βˆ’7π‘₯) = 7 Γ— 7 (500γ€– 𝑒〗^7π‘₯+600γ€– 𝑒〗^(βˆ’7π‘₯) ) = 49 (500γ€– 𝑒〗^7π‘₯+600γ€– 𝑒〗^(βˆ’7π‘₯) ) = 49 𝑦 Hence proved .

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