Check Full Chapter Explained - Continuity and Differentiability - https://you.tube/Chapter-5-Class-12-Continuity

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise

Transcript

Example 29 Differentiate the following w.r.t. x: (i) 𝑒^(–π‘₯) Let 𝑦 = 𝑒^(–π‘₯) Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑(𝑦)/𝑑π‘₯ = 𝑑(𝑒^(βˆ’π‘₯) )/𝑑π‘₯ 𝑑(𝑦)/𝑑π‘₯ = 𝑒^(βˆ’π‘₯) . 𝑑(βˆ’π‘₯)/𝑑π‘₯ 𝑑(𝑦)/𝑑π‘₯ = 𝑒^(βˆ’π‘₯) (βˆ’1) . 𝒅(π’š)/𝒅𝒙 = γ€–βˆ’π’†γ€—^(βˆ’π’™) Example 29 Differentiate the following w.r.t. x: (ii) sin⁑(log⁑π‘₯), π‘₯ > 0 Let 𝑦 =sin⁑(log⁑π‘₯) Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑𝑦/𝑑π‘₯ = (𝑑(sin⁑(log⁑π‘₯)) )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = cos⁑(log⁑π‘₯) . (𝑑⁑(log⁑π‘₯ ) )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = cos⁑(log⁑π‘₯) . 1/π‘₯ π’…π’š/𝒅𝒙 = (𝒄𝒐𝒔⁑(π’π’π’ˆβ‘π’™)" " )/𝒙 (𝑠𝑖𝑛⁑π‘₯ )^β€²=π‘π‘œπ‘ β‘π‘₯ ((π‘™π‘œπ‘”β‘π‘₯ )^β€²= 1/π‘₯) Example 29 Differentiate the following w.r.t. x: (iii) γ€–π‘π‘œπ‘ γ€—^(βˆ’1) "(ex)" Let 𝑦 = γ€–π‘π‘œπ‘ γ€—^(βˆ’1) "(ex)" Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑑(cos^(βˆ’1)⁑(𝑒^π‘₯ ) )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = (βˆ’1)/√(1 βˆ’ (𝑒^π‘₯ )^2 ) . 𝑑(𝑒^π‘₯ )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = (βˆ’1)/√(1 βˆ’ (𝑒^π‘₯ )^2 ) . 𝑒^π‘₯ π’…π’š/𝒅𝒙 = (βˆ’π’†^𝒙)/√(𝟏 βˆ’ 𝒆^πŸπ’™ ) (As (cos^(βˆ’1)⁑π‘₯ )^β€²=(βˆ’1)/√(1 βˆ’ π‘₯^2 )) (As(𝑒^π‘₯ )^β€²=𝑒^π‘₯) Example 29 Differentiate the following w.r.t. x: (iv) π‘’π‘π‘œπ‘  π‘₯ Let 𝑦 = 𝑒^cos⁑π‘₯ Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑑(𝑒^cos⁑π‘₯ )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑒^cos⁑π‘₯ . 𝑑(cos⁑π‘₯ )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑒^cos⁑π‘₯ . (βˆ’sin⁑π‘₯ ) π’…π’š/𝒅𝒙 = γ€–βˆ’π’”π’Šπ’β‘π’™. 𝒆〗^𝒄𝒐𝒔⁑𝒙 ((𝑒^π‘₯)β€²=𝑒^π‘₯) ((cos⁑〖π‘₯)β€²=βˆ’sin⁑π‘₯ γ€— )

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.