Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12




Last updated at March 11, 2021 by Teachoo
Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12
Transcript
Ex 5.5, 17 Differentiate (๐ฅ^2 โ 5๐ฅ + 8) (๐ฅ^3 + 7๐ฅ + 9) (ii) by expanding the product to obtain a single polynomial.By Expanding the product to obtain a single polynomial . ๐ฆ=(๐ฅ^2 " โ 5" ๐ฅ" + 8" ) (๐ฅ^3 " + 7" ๐ฅ" + 9" ) ๐ฆ=๐ฅ^2 (๐ฅ^3 " + 7" ๐ฅ" + 9" )" โ 5" ๐ฅ(๐ฅ^3 " + 7" ๐ฅ" + 9" )" + 8 " (๐ฅ^3 " + 7" ๐ฅ" + 9" ) ๐ฆ=๐ฅ^5+7๐ฅ^3+9๐ฅ^2โ5๐ฅ^4โ35๐ฅ^2โ45๐ฅ+8๐ฅ^3+56๐ฅ+72 ๐ฆ=๐ฅ^5โ5๐ฅ^4+15๐ฅ^3โ26๐ฅ^2+11๐ฅ+72 Differentiating both sides ๐ค.๐.๐ก.๐ฅ. ๐๐ฆ/๐๐ฅ = (๐(๐ฅ^5 โ 5๐ฅ^4 + 15๐ฅ^3โ 26๐ฅ^2 + 11๐ฅ + 72" " )" " )/๐๐ฅ ๐๐ฆ/๐๐ฅ = (๐(๐ฅ^5))/๐๐ฅ โ (๐(5๐ฅ^4))/๐๐ฅ + (๐(15๐ฅ^3)" " )/๐๐ฅ โ (๐(26๐ฅ^2)" " )/๐๐ฅ + (๐(11๐ฅ)" " )/๐๐ฅ + (๐(72)" " )/๐๐ฅ ๐๐ฆ/๐๐ฅ = 5๐ฅ^4โ20๐ฅ^3+45๐ฅ^2โ52๐ฅ+11 + 0 ๐ ๐/๐ ๐ = ๐๐^๐โ๐๐๐^๐+๐๐๐^๐โ๐๐๐+๐๐ Ex 5.5, 17 Differentiate (๐ฅ^2โ 5 ๐ฅ + 8) (๐ฅ^3 + 7 ๐ฅ + 9) (iii) by logarithmic differentiation.By logarithmic differentiation ๐ฆ= (๐ฅ^2 "โ 5 " ๐ฅ" + 8" ) (๐ฅ^3 " + 7 " ๐ฅ" + 9" ) Taking log both sides log ๐ฆ = log ((๐ฅ^2 " โ 5" ๐ฅ" + 8" ) (๐ฅ^3 " + 7" ๐ฅ" + 9" )) log ๐ฆ=log (๐ฅ^2 " โ 5" ๐ฅ" + 8" )+ใlog ใโก(๐ฅ^3 " + 7" ๐ฅ" + 9" ) Differentiating both sides ๐ค.๐.๐ก.๐ฅ. (๐(logโก๐ฆ ) )/๐๐ฅ = ๐(log (๐ฅ^2 " โ " 5๐ฅ" + " 8) + ใlog ใโก(๐ฅ^3 " + " 7๐ฅ" +" 9) )/๐๐ฅ (๐(logโก๐ฆ ) )/๐๐ฅ . ๐๐ฆ/๐๐ฆ = ๐(log (๐ฅ^2 " โ " 5๐ฅ" + " 8))/๐๐ฅ + ๐(ใlog ใโก(๐ฅ^3 " + " 7๐ฅ" +" 9) )/๐๐ฅ (๐(logโก๐ฆ ) )/๐๐ฆ . ๐๐ฆ/๐๐ฅ = 1/((๐ฅ^2 " โ " 5๐ฅ" + " 8) ) . ๐(๐ฅ^2 " โ " 5๐ฅ" + " 8)/๐๐ฅ + 1/((๐ฅ^3 " + " 7๐ฅ" +" 9) ) . ๐(๐ฅ^3 " + " 7๐ฅ" +" 9)/๐๐ฅ (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = 1/(๐ฅ^2 " โ " 5๐ฅ" + " 8) . (2x โ 5 + 0) + 1/(๐ฅ^3 " + " 7๐ฅ" +" 9) .(3x2 + 7 + 0) (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = ((2๐ฅ โ 5))/(๐ฅ^2 " โ " 5๐ฅ" + " 8) + ((3๐ฅ^2 + 7))/(๐ฅ^3 " + " 7๐ฅ" +" 9) (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = ((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" +" 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) ) ๐๐ฆ/๐๐ฅ = ๐ฆ(((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" +" 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) )) ๐๐ฆ/๐๐ฅ =(๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9)(((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" + " 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) )) (๐(logโก๐ฆ ) )/๐๐ฅ . ๐๐ฆ/๐๐ฆ = ๐(log (๐ฅ^2 " โ " 5๐ฅ" + " 8))/๐๐ฅ + ๐(ใlog ใโก(๐ฅ^3 " + " 7๐ฅ" +" 9) )/๐๐ฅ (๐(logโก๐ฆ ) )/๐๐ฆ . ๐๐ฆ/๐๐ฅ = 1/((๐ฅ^2 " โ " 5๐ฅ" + " 8) ) . ๐(๐ฅ^2 " โ " 5๐ฅ" + " 8)/๐๐ฅ + 1/((๐ฅ^3 " + " 7๐ฅ" +" 9) ) . ๐(๐ฅ^3 " + " 7๐ฅ" +" 9)/๐๐ฅ (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = 1/(๐ฅ^2 " โ " 5๐ฅ" + " 8) . (2x โ 5 + 0) + 1/(๐ฅ^3 " + " 7๐ฅ" +" 9) .(3x2 + 7 + 0) (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = ((2๐ฅ โ 5))/(๐ฅ^2 " โ " 5๐ฅ" + " 8) + ((3๐ฅ^2 + 7))/(๐ฅ^3 " + " 7๐ฅ" +" 9) (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = ((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" +" 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) ) ๐๐ฆ/๐๐ฅ = ๐ฆ(((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" +" 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) )) ๐๐ฆ/๐๐ฅ =(๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9)(((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" + " 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) )) (๐(logโก๐ฆ ) )/๐๐ฅ . ๐๐ฆ/๐๐ฆ = ๐(log (๐ฅ^2 " โ " 5๐ฅ" + " 8))/๐๐ฅ + ๐(ใlog ใโก(๐ฅ^3 " + " 7๐ฅ" +" 9) )/๐๐ฅ (๐(logโก๐ฆ ) )/๐๐ฆ . ๐๐ฆ/๐๐ฅ = 1/((๐ฅ^2 " โ " 5๐ฅ" + " 8) ) . ๐(๐ฅ^2 " โ " 5๐ฅ" + " 8)/๐๐ฅ + 1/((๐ฅ^3 " + " 7๐ฅ" +" 9) ) . ๐(๐ฅ^3 " + " 7๐ฅ" +" 9)/๐๐ฅ (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = 1/(๐ฅ^2 " โ " 5๐ฅ" + " 8) . (2x โ 5 + 0) + 1/(๐ฅ^3 " + " 7๐ฅ" +" 9) .(3x2 + 7 + 0) (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = ((2๐ฅ โ 5))/(๐ฅ^2 " โ " 5๐ฅ" + " 8) + ((3๐ฅ^2 + 7))/(๐ฅ^3 " + " 7๐ฅ" +" 9) (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = ((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" +" 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) ) ๐๐ฆ/๐๐ฅ = ๐ฆ(((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" +" 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) )) ๐๐ฆ/๐๐ฅ =(๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9)(((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" + " 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) )) (๐(logโก๐ฆ ) )/๐๐ฅ . ๐๐ฆ/๐๐ฆ = ๐(log (๐ฅ^2 " โ " 5๐ฅ" + " 8))/๐๐ฅ + ๐(ใlog ใโก(๐ฅ^3 " + " 7๐ฅ" +" 9) )/๐๐ฅ (๐(logโก๐ฆ ) )/๐๐ฆ . ๐๐ฆ/๐๐ฅ = 1/((๐ฅ^2 " โ " 5๐ฅ" + " 8) ) . ๐(๐ฅ^2 " โ " 5๐ฅ" + " 8)/๐๐ฅ + 1/((๐ฅ^3 " + " 7๐ฅ" +" 9) ) . ๐(๐ฅ^3 " + " 7๐ฅ" +" 9)/๐๐ฅ (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = 1/(๐ฅ^2 " โ " 5๐ฅ" + " 8) . (2x โ 5 + 0) + 1/(๐ฅ^3 " + " 7๐ฅ" +" 9) .(3x2 + 7 + 0) (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = ((2๐ฅ โ 5))/(๐ฅ^2 " โ " 5๐ฅ" + " 8) + ((3๐ฅ^2 + 7))/(๐ฅ^3 " + " 7๐ฅ" +" 9) (1 )/๐ฆ . ๐๐ฆ/๐๐ฅ = ((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" +" 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) ) ๐๐ฆ/๐๐ฅ = ๐ฆ(((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" +" 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) )) ๐๐ฆ/๐๐ฅ =(๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9)(((2๐ฅ โ 5) (๐ฅ^3 " + " 7๐ฅ" + " 9) + (3๐ฅ^2 + 7) (๐ฅ^2 " โ " 5๐ฅ" + " 8))/((๐ฅ^2 " โ " 5๐ฅ" + " 8) (๐ฅ^3 " + " 7๐ฅ" +" 9) ))
Ex 5.5
Ex 5.5, 2
Ex 5.5, 3 Important
Ex 5.5, 4
Ex 5.5, 5
Ex 5.5,6 Important
Ex 5.5, 7 Important
Ex 5.5, 8
Ex 5.5, 9 Important
Ex 5.5, 10 Important
Ex 5.5, 11 Important
Ex 5.5, 12
Ex 5.5, 13
Ex 5.5, 14 Important
Ex 5.5, 15
Ex 5.5, 16 Important
Ex 5.5, 17 Important You are here
Ex 5.5, 18
About the Author