Q.3 (A) (iii) If x⁵· y⁷=(x+y)¹² then show that dy/dx=y/x.

Answer: Proved below. Taking logarithm: 5log x+7log y=12log(x+y). Differentiating w.r.t. x: 5/x+7/ydy/dx=12/(x+y)(1+dy/dx) therefore(7/y-12/(x+y))dy/dx=12/(x+y)-5/x therefore(7x+7y-12y)/(y(x+y))·dy/dx=(12x-5x-5y)/(x(x+y)) therefore(7x-5y)/(y(x+y))·dy/dx=(7x-5y)/(x(x+y)) thereforedy/dx=y/x. Hence proved.

Chapter: Differentiation· Mathematics & Statistics (Commerce) Part 1

If \(x^5\cdot y^7=(x+y)^{12}\) then show that \(\frac{dy}{dx}=\frac{y}{x}\).

Q.3 (A) (iii) If \(x^5\cdot y^7=(x+y)^{12}\) then show that \(\frac{dy}{dx}=\frac{y}{x}\).

Doubtly Asked 29 Oct 2026 Board exam · July 2025 hsc 12th Save WhatsApp Report

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Answer: Proved below.

Taking logarithm: \(5\log x+7\log y=12\log(x+y)\).

Differentiating w.r.t. \(x\): \(\frac{5}{x}+\frac{7}{y}\frac{dy}{dx}=\frac{12}{x+y}\left(1+\frac{dy}{dx}\right)\)

\(\therefore\left(\frac{7}{y}-\frac{12}{x+y}\right)\frac{dy}{dx}=\frac{12}{x+y}-\frac{5}{x}\)
\(\therefore\frac{7x+7y-12y}{y(x+y)}\cdot\frac{dy}{dx}=\frac{12x-5x-5y}{x(x+y)}\)
\(\therefore\frac{7x-5y}{y(x+y)}\cdot\frac{dy}{dx}=\frac{7x-5y}{x(x+y)}\)

\(\therefore\frac{dy}{dx}=\frac{y}{x}\). Hence proved.

Doubtly AI · 29 Oct 2026 AI-verified · Nov 2026 Report

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If \(x^5\cdot y^7=(x+y)^{12}\) then show that \(\frac{dy}{dx}=\frac{y}{x}\).

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