Logarithmic Differentiation - Type 1

Chapter 5 Class 12 Continuity and Differentiability
Concept wise

### Transcript

Example 29 Differentiate π₯^sinβ‘π₯ , π₯ > 0 π€.π.π‘. π₯.Let y = π₯^sinβ‘π₯ Taking log both sides logβ‘π¦ = log π₯^sinβ‘π₯ πππβ‘π = πππβ‘π . πππ π Differentiating π€.π.π‘.π₯ (π(logβ‘γπ¦)γ)/ππ₯ = π/ππ₯ (sinβ‘γπ₯ logβ‘π₯ γ ) By product Rule (uv)β = uβv + vβu where u = sin x & v = log x (π(logβ‘γπ¦)γ)/ππ₯ = (π(sinβ‘π₯))/ππ₯.log π₯+sin π₯ . (π(logβ‘π₯))/ππ₯ (π(logβ‘γπ¦)γ)/ππ¦ Γ ππ¦/ππ₯ = cosβ‘π₯ logβ‘π₯ + sinβ‘π₯ 1/π₯ ππ¦/ππ₯ 1/π¦ = πππ πβ‘πππβ‘π + πππβ‘π π/π ππ¦/ππ₯ = π¦ (γπππ  π₯γβ‘γπππβ‘γπ₯+1/π₯γ π ππβ‘π₯ γ ) Putting back π¦ = π₯^π ππβ‘π₯ ππ¦/ππ₯ = π₯^π ππβ‘π₯ (cosβ‘γlogβ‘γπ₯+ 1/π₯γ sinβ‘π₯ γ ) = π₯^π ππβ‘π₯ cosβ‘logβ‘π₯ + π₯^π ππβ‘π₯ 1/π₯ π ππβ‘π₯ = π₯^π ππβ‘π₯ cosβ‘logβ‘π₯ + π₯^π ππβ‘π₯ π₯^(β1) sinβ‘π₯ = π^πππβ‘π πππβ‘πππβ‘π + π^πππβ‘γπ β πγ πππβ‘π

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.