Ex 12.2

Chapter 12 Class 11 Limits and Derivatives
Serial order wise

### Transcript

Ex 12.2, 4 Find the derivative of the following functions from first principle. (iii) 1/๐ฅ^2 Let f (x) = 1/๐ฅ^2 We need to find derivative of f(x) i.e. fโ (x) We know that fโ(x) = (๐๐๐)โฌ(โโ0) ๐โกใ(๐ฅ + โ) โ ๐(๐ฅ)ใ/โ Here, f (x) = 1/๐ฅ^2 So, f (x + h) = 1/ใ(๐ฅ + โ)ใ^2 Putting values fโ(x) = (๐๐๐)โฌ(โโ0)โกใ(1/ใ(๐ฅ + โ)ใ^2 โ 1/๐ฅ^2 )/โใ = (๐๐๐)โฌ(โโ0)โกใ(ใใ๐ฅ ใ^2 โ (๐ฅ + โ)ใ^2/(ใ(๐ฅ + โ)ใ^2 ๐ฅ^2 ))/โใ = (๐๐๐)โฌ(โโ0)โกใ( ๐ฅ2 โ (๐ฅ + โ)2)/(โ๐ฅ2 (๐ฅ + โ)2)ใ = (๐๐๐)โฌ(โโ0)โกใ((๐ฅ โ ( ๐ฅ + โ )) (๐ฅ + (๐ฅ + โ)))/(โ๐ฅ2 (๐ฅ + โ)2)ใ = (๐๐๐)โฌ(โโ0)โกใ( (๐ฅ โ๐ฅ โ โ) (๐ฅ + ๐ฅ + โ))/(โ.๐ฅ2 (๐ฅ + โ)2)ใ (As a2 โ b2 = (a โ b) (a + b)) = (๐๐๐)โฌ(โโ0)โกใ((โโ)(2๐ฅ + โ))/(โ๐ฅ2 (๐ฅ + โ)2)ใ = (๐๐๐)โฌ(โโ0)โกใ((โ1) 2๐ฅ + โ)/(๐ฅ2 (๐ฅ + โ)2)ใ Putting h = 0 = ((โ1) 2๐ฅ + 0)/(๐ฅ2 (๐ฅ + 0)2) = ((โ1) 2๐ฅ)/(๐ฅ2 (๐ฅ)2) = (โ2๐ฅ)/๐ฅ4 = (โ2)/๐ฅ^3 Thus, fโ(x) = (โ๐)/๐^๐