Q.21 Find the n^th order derivative of log x .

Answer: dⁿy/dxⁿ = dfrac(-1)ⁿ⁻¹(n - 1)!xⁿ Let y = log x . Then y₁ = 1/x = x⁻¹ , y₂ = (-1)x⁻² = (-1)¹ 1!/x² , y₃ = (-1)(-2)x⁻³ = (-1)² 2!/x³ , y₄ = (-1)³ 3!/x⁴ , … Following the pattern, y_n = dfrac(-1)ⁿ⁻¹(n - 1)!xⁿ .

Find the \(n^{\text{th}}\) order derivative of \(\log x\).

Q.21 Find the \(n^{\text{th}}\) order derivative of \(\log x\).

Doubtly Asked 28 Oct 2026 Board exam · March 2025 hsc 12th Save WhatsApp Report

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Answer: \(\dfrac{d^ny}{dx^n} = \dfrac{(-1)^{n-1}(n - 1)!}{x^n}\)

Let \(y = \log x\). Then

\(y_1 = \dfrac1x = x^{-1}\), \(y_2 = (-1)x^{-2} = \dfrac{(-1)^1\,1!}{x^2}\), \(y_3 = (-1)(-2)x^{-3} = \dfrac{(-1)^2\,2!}{x^3}\), \(y_4 = \dfrac{(-1)^3\,3!}{x^4}\), …

Following the pattern, \(y_n = \dfrac{(-1)^{n-1}(n - 1)!}{x^n}\).

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report

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Find the \(n^{\text{th}}\) order derivative of \(\log x\).

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