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How do we differentiate a^x? - Example 31 - Chapter 5 Class 12

Example 31 - Chapter 5 Class 12 Continuity and Differentiability - Part 2


Transcript

Example 31 Differentiate π‘Ž^π‘₯ 𝑀.π‘Ÿ.𝑑.π‘₯, where a is a positive constant.Let y = π‘Ž^π‘₯ Taking log on both sides log⁑𝑦 = logβ‘π‘Ž^π‘₯ π’π’π’ˆβ‘π’š = 𝒙 π’π’π’ˆβ‘ 𝒂 Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ (𝑑(log⁑𝑦))/𝑑π‘₯ = 𝑑/𝑑π‘₯(π‘₯ logβ‘π‘Ž) (𝑑(log⁑𝑦))/𝑑π‘₯ = logβ‘π‘Ž (𝑑π‘₯/𝑑π‘₯) (𝑑(log⁑𝑦))/𝑑π‘₯ = π’π’π’ˆβ‘π’‚ (π‘™π‘œπ‘”β‘γ€–π‘Ž^𝑏=𝑏 π‘™π‘œπ‘”β‘π‘Ž γ€—) (𝑑(log⁑𝑦))/𝑑π‘₯ . 𝑑𝑦/𝑑𝑦 = logβ‘π‘Ž (𝑑(log⁑𝑦))/𝑑𝑦 . 𝑑𝑦/𝑑π‘₯ = logβ‘π‘Ž 1/𝑦 . 𝑑𝑦/𝑑π‘₯ = logβ‘π‘Ž 𝑑𝑦/𝑑π‘₯ = 𝑦 logβ‘π‘Ž Putting back 𝑦 = π‘Ž^π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝒂^𝒙 π’π’π’ˆβ‘π’‚

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.