Ex 5.5, 5 - Differentiate (x + 3)^2 (x + 4)^3 (x + 5)^4 - Teachoo

Ex 5.5, 5 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.5, 5 - Chapter 5 Class 12 Continuity and Differentiability - Part 3 Ex 5.5, 5 - Chapter 5 Class 12 Continuity and Differentiability - Part 4

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Transcript

Ex 5.5, 5 Differentiate the functions in, (𝑥 + 3)^2 . (𝑥 + 4)^3 . (𝑥 + 5)^4 Let 𝑦= (𝑥 + 3)^2 . (𝑥 + 4)^3 . (𝑥 + 5)^4 Taking log both sides log⁡𝑦 = log ((𝑥 + 3)^2 . (𝑥 + 4)^3 . (𝑥 + 5)^4 ) log⁡𝑦 = log (𝑥 + 3)^2 + log (𝑥 + 4)^3 + log (𝑥 + 5)^4 log⁡𝑦 = 2 log (𝑥 + 3) + 3 log (𝑥 + 4) + 4 log (𝑥 + 5) Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑(log⁡𝑦 )/𝑑𝑥 = (𝑑 (2 log (𝑥 + 3)" + " 3 log (𝑥 + 4)" + " 4 log (𝑥 + 5)))/𝑑𝑥 𝑑(log⁡𝑦 )/𝑑𝑥 (𝑑𝑦/𝑑𝑦) = 𝑑(2 log (𝑥 + 3))/𝑑𝑥 + (𝑑 (3 log (𝑥 + 4)))/𝑑𝑥 + 𝑑(4 log (𝑥 + 5))/𝑑𝑥 𝑑(log⁡𝑦 )/𝑑𝑦 (𝑑𝑦/𝑑𝑥) = 2 𝑑(log (𝑥 + 3))/𝑑𝑥 + 3 (𝑑 (log (𝑥 + 4)))/𝑑𝑥 + 4 𝑑(log (𝑥 + 5))/𝑑𝑥 1/𝑦 × 𝑑𝑦/𝑑𝑥 = 2. 1/((𝑥 + 3) ) . 𝑑(𝑥 + 3)/𝑑𝑥 + 3. 1/((𝑥 + 4) ) . 𝑑(𝑥 + 4)/𝑑𝑥 + 4. 1/((𝑥 + 5) ) . 𝑑(𝑥 + 5)/𝑑𝑥 1/𝑦 × 𝑑𝑦/𝑑𝑥 = 2/(𝑥 + 3) (𝑑𝑥/𝑑𝑥+𝑑(3)/𝑑𝑥) + 3/(𝑥 + 4) (𝑑𝑥/𝑑𝑥+𝑑(4)/𝑑𝑥) + 4/(𝑥 +5) (𝑑𝑥/𝑑𝑥+𝑑(5)/𝑑𝑥) 1/𝑦 × 𝑑𝑦/𝑑𝑥 = 2/(𝑥 + 3) (1+0) + 3/(𝑥 + 4) (1+0) + 4/(𝑥 + 5) (1+0) 1/𝑦 × 𝑑𝑦/𝑑𝑥 = 2/(𝑥 + 3) + 3/(𝑥 + 4) + 4/(𝑥 + 5) 𝑑𝑦/𝑑𝑥 = 𝑦 (2/(𝑥 + 3) " + " 3/(𝑥 + 4) " + " 4/(𝑥 + 5)) Putting value of 𝑦 𝑑𝑦/𝑑𝑥 = (𝑥 + 3)^2 . (𝑥 + 4)^3 . (𝑥 + 5)^(4 ) (2/((𝑥 + 3) ) "+ " 3/((𝑥 + 4) ) " + " 4/((𝑥 + 5) )) 𝑑𝑦/𝑑𝑥 = (𝑥 + 3)^2 (𝑥 + 4)^3 (𝑥 + 5)^(4 ) ((2(𝑥 + 4) (𝑥 + 5) + 3(𝑥 + 3) (𝑥 + 5) + 4(𝑥 + 3) (𝑥 + 4))/((𝑥 + 3) (𝑥 + 4) (𝑥 + 5) )) 𝑑𝑦/𝑑𝑥 = ((𝑥 + 3)^2 (𝑥 + 4)^3 〖 (𝑥 + 5)〗^(4 ))/((𝑥 + 3) (𝑥 + 4) (𝑥 + 5) ) (2(𝑥^2+4𝑥+5𝑥+20)+3(𝑥^2+3𝑥+5𝑥+15)+ 4(𝑥^2+3𝑥+4𝑥+12)) 𝑑𝑦/𝑑𝑥 =(𝑥 + 3) (𝑥 + 4)^2 〖 (𝑥 + 5)〗^(3 ) (2(𝑥^2+9𝑥+20)+3(𝑥^2+8𝑥+15)+4(𝑥^2+7𝑥+12)) 𝑑𝑦/𝑑𝑥 =(𝑥 + 3) (𝑥 + 4)^2 〖 (𝑥 + 5)〗^(3 ) (2𝑥^2+18𝑥+40+3𝑥^2+24𝑥+45+4𝑥^2+28𝑥+48) 𝑑𝑦/𝑑𝑥 =(𝑥 + 3) (𝑥 + 4)^2 〖 (𝑥 + 5)〗^(3 ) (2𝑥^2+3𝑥^2+4𝑥^2 18𝑥+24𝑥+28𝑥+40+45+48) 𝒅𝒚/𝒅𝒙 =(𝒙 + 𝟑) (𝒙 + 𝟒)^𝟐 〖 (𝒙 + 𝟓)〗^(𝟑 ) (𝟗𝒙^𝟐+𝟕𝟎𝒙+𝟏𝟑𝟑)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.