Q.21 If y = log[x + √(x² + a²)/√(x² + a²) - x] , find dy/dx .

Answer: dy/dx = 2/√(x² + a²) Rationalise: x + √(x² + a²)/√(x² + a²) - x×√(x² + a²) + x/√(x² + a²) + x = (x + √(x² + a²))²/a² . So y = 2log(x + √(x² + a²)) - 2log a . dy/dx = 2/x + √(x² + a²)(1 + x/√(x² + a²)) = 2/x + √(x² + a²)·√(x² + a²) + x/√(x² + a²) = 2/√(x² + a²) .

If \(y = \log\left[\frac{x + \sqrt{x^2 + a^2}}{\sqrt{x^2 + a^2} - x}\right]\), find \(\frac{dy}{dx}\).

Q.21 If \(y = \log\left[\frac{x + \sqrt{x^2 + a^2}}{\sqrt{x^2 + a^2} - x}\right]\), find \(\frac{dy}{dx}\).

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

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Answer: \(\frac{dy}{dx} = \frac{2}{\sqrt{x^2 + a^2}}\)

Rationalise: \(\frac{x + \sqrt{x^2 + a^2}}{\sqrt{x^2 + a^2} - x}\times\frac{\sqrt{x^2 + a^2} + x}{\sqrt{x^2 + a^2} + x} = \frac{(x + \sqrt{x^2 + a^2})^2}{a^2}\).

So \(y = 2\log\left(x + \sqrt{x^2 + a^2}\right) - 2\log a\).

\(\frac{dy}{dx} = \frac{2}{x + \sqrt{x^2 + a^2}}\left(1 + \frac{x}{\sqrt{x^2 + a^2}}\right) = \frac{2}{x + \sqrt{x^2 + a^2}}\cdot\frac{\sqrt{x^2 + a^2} + x}{\sqrt{x^2 + a^2}} = \frac{2}{\sqrt{x^2 + a^2}}\).

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

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If \(y = \log\left[\frac{x + \sqrt{x^2 + a^2}}{\sqrt{x^2 + a^2} - x}\right]\), find \(\frac{dy}{dx}\).

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