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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

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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 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.