





Last updated at May 29, 2018 by Teachoo
Transcript
Ex 5.5, 17 Differentiate ( 𝑥2 – 5𝑥 + 8) ( 𝑥3 + 7𝑥 + 9) (𝑖) By using product rule Let y = ( 𝑥2 – 5𝑥 + 8) ( 𝑥3 + 7𝑥 + 9) By using product rule 𝑑𝑦𝑑𝑥= 𝑑 𝑥2 − 5𝑥 + 8 𝑥3 + 7𝑥 + 9𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑 𝑥2 – 5𝑥 + 8 𝑑𝑥 . 𝑥3+7𝑥+9 + 𝑑 𝑥3+ 7𝑥 + 9𝑑𝑥 . 𝑥2 – 5𝑥 + 8 𝑑𝑦𝑑𝑥 = 2𝑥−5+0 𝑥3+7𝑥+9 + 3 𝑥2+7+0 𝑥2 – 5𝑥 + 8 𝑑𝑦𝑑𝑥 = 2𝑥−5 𝑥3+7𝑥+9 + 3 𝑥2+7 𝑥2 – 5𝑥 + 8 𝑑𝑦𝑑𝑥 = 2𝑥 𝑥3+7𝑥+9−5 𝑥3+7𝑥+9 + 3 𝑥2 𝑥2 – 5𝑥 + 8 + 7 𝑥2 – 5𝑥 + 8 𝑑𝑦𝑑𝑥 = 2 𝑥4+14 𝑥2+18𝑥−5 𝑥3 − 35𝑥−45+3 𝑥4 −15𝑥 + 24 𝑥2+7 𝑥2−35𝑥+5 𝒅𝒚𝒅𝒙 = 𝟓 𝒙𝟒−𝟐𝟎 𝒙𝟑+ 𝟒𝟓 𝒙𝟐−𝟓𝟐𝒙+𝟏𝟏 Ex 5.5, 17 Differentiate ( 𝑥2– 5 𝑥 + 8) ( 𝑥3 + 7 𝑥 + 9) (ii) by expanding the product to obtain a single polynomial. By Expanding the product to obtain a single polynomial . 𝑦= 𝑥2– 5 𝑥 + 8 𝑥3 + 7 𝑥 + 9 𝑦= 𝑥2 𝑥3 + 7 𝑥 + 9 – 5𝑥 𝑥3 + 7 𝑥 + 9 8 𝑥3 + 7 𝑥 + 9 𝑦= 𝑥5+7 𝑥3+9 𝑥2−5 𝑥4−35 𝑥2−45𝑥+8 𝑥3+56𝑥+72 𝑦= 𝑥5−5 𝑥4+15 𝑥3−26 𝑥2+11𝑥+72 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑𝑦𝑑𝑥 = 𝑑( 𝑥5 − 5 𝑥4 + 15 𝑥3− 26 𝑥2 + 11𝑥 + 72 ) 𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝑑( 𝑥5)𝑑𝑥 − 𝑑(5 𝑥4)𝑑𝑥 + 𝑑(15 𝑥3) 𝑑𝑥 − 𝑑(26 𝑥2) 𝑑𝑥 + 𝑑(11𝑥) 𝑑𝑥 + 𝑑(72) 𝑑𝑥 𝑑𝑦𝑑𝑥 = 5 𝑥4−20 𝑥3+45 𝑥2−52𝑥+11 + 0 𝒅𝒚𝒅𝒙 = 𝟓 𝒙𝟒−𝟐𝟎 𝒙𝟑+𝟒𝟓 𝒙𝟐−𝟓𝟐𝒙+𝟏𝟏 Ex 5.5, 17 Differentiate ( 𝑥2– 5 𝑥 + 8) ( 𝑥3 + 7 𝑥 + 9) (iii) by logarithmic differentiation. By logarithmic differentiation 𝑦= 𝑥2– 5 𝑥 + 8 𝑥3 + 7 𝑥 + 9 Taking log both sides log 𝑦 = log 𝑥2– 5 𝑥 + 8 𝑥3 + 7 𝑥 + 9 log 𝑦=log 𝑥2– 5 𝑥 + 8+ log 𝑥3 + 7 𝑥 + 9 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑 log𝑦 𝑑𝑥 = 𝑑 log 𝑥2 – 5𝑥 + 8 + log 𝑥3 + 7𝑥 + 9𝑑𝑥 𝑑 log𝑦 𝑑𝑥 . 𝑑𝑦𝑑𝑦 = 𝑑 log 𝑥2 – 5𝑥 + 8𝑑𝑥 + 𝑑 log 𝑥3 + 7𝑥 + 9𝑑𝑥 𝑑 log𝑦 𝑑𝑦 . 𝑑𝑦𝑑𝑥 = 1 𝑥2 – 5𝑥 + 8 . 𝑑 𝑥2 – 5𝑥 + 8𝑑𝑥 + 1 𝑥3 + 7𝑥 + 9 . 𝑑 𝑥3 + 7𝑥 + 9𝑑𝑥 1 𝑦 . 𝑑𝑦𝑑𝑥 = 1 𝑥2 – 5𝑥 + 8 . (2x – 5 + 0) + 1 𝑥3 + 7𝑥 + 9 .(3x2 + 7 + 0) 1 𝑦 . 𝑑𝑦𝑑𝑥 = 2𝑥 − 5 𝑥2 – 5𝑥 + 8 + 3 𝑥2 + 7 𝑥3 + 7𝑥 + 9 1 𝑦 . 𝑑𝑦𝑑𝑥 = 2𝑥 − 5 𝑥3 + 7𝑥 + 9 + 3 𝑥2 + 7 𝑥2 – 5𝑥 + 8 𝑥2 – 5𝑥 + 8 𝑥3 + 7𝑥 + 9 𝑑𝑦𝑑𝑥 = 𝑦 2𝑥 − 5 𝑥3 + 7𝑥 + 9 + 3 𝑥2 + 7 𝑥2 – 5𝑥 + 8 𝑥2 – 5𝑥 + 8 𝑥3 + 7𝑥 + 9 𝑑𝑦𝑑𝑥 = 𝑥2 – 5𝑥 + 8 𝑥3 + 7𝑥 + 9 2𝑥 − 5 𝑥3 + 7𝑥 + 9 + 3 𝑥2 + 7 𝑥2 – 5𝑥 + 8 𝑥2 – 5𝑥 + 8 𝑥3 + 7𝑥 + 9 𝑑𝑦𝑑𝑥 = 2𝑥−5 𝑥3+7𝑥+9 + 3 𝑥2+7 𝑥2 – 5𝑥 + 8 𝑑𝑦𝑑𝑥 = 2𝑥 𝑥3+7𝑥+9−5 𝑥3+7𝑥+9 + 3 𝑥2 𝑥2 – 5𝑥 + 8 + 7 𝑥2 – 5𝑥 + 8 𝑑𝑦𝑑𝑥 = 2 𝑥4+14 𝑥2+18𝑥−5 𝑥3 − 35𝑥−45+3 𝑥4 −15𝑥 + 24 𝑥2+7 𝑥2−35𝑥+5 𝒅𝒚𝒅𝒙 = 𝟓 𝒙𝟒−𝟐𝟎 𝒙𝟑+ 𝟒𝟓 𝒙𝟐−𝟓𝟐𝒙+𝟏𝟏 Hence, the answer is same in all three cases .
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