Q.1 (v) Let f(1) = 3 , f'(1) = -1/3 , g(1) = -4 and g'(1) = -8/3 . The derivative of √([f(x)]² + [g(x)]²) w.r.t. x at x = 1 is _____. A. -29/25 B. 7/3 C. 31/15 D. 29/15

Answer: D. 29/15 Answer: (d) 29/15 Let h(x) = √(f² + g²) . Then h'(x) = f(x)f'(x) + g(x)g'(x)/√(f² + g²) . At x = 1 : √(9 + 16) = 5 , f f' = 3(-1/3) = -1 , g g' = (-4)(-8/3) = 32/3 . h'(1) = -1 + 32/3/5 = 29/3/5 = 29/15 .

Let \(f(1) = 3\), \(f'(1) = -\dfrac{1}{3}\), \(g(1) = -4\) and \(g'(1) = -\dfrac{8}{3}\). The derivative of \(\sqrt{[f(x)]^2 + [g(x)]^2}\) w.r.t. \(x\) at \(x = 1\) is _____.

Q.1 (v) Let \(f(1) = 3\), \(f'(1) = -\dfrac{1}{3}\), \(g(1) = -4\) and \(g'(1) = -\dfrac{8}{3}\). The derivative of \(\sqrt{[f(x)]^2 + [g(x)]^2}\) w.r.t. \(x\) at \(x = 1\) is _____.

A. \(-\dfrac{29}{25}\)

B. \(\dfrac{7}{3}\)

C. \(\dfrac{31}{15}\)

D. \(\dfrac{29}{15}\)

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

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Answer: D. \(\dfrac{29}{15}\)

Answer: (d) \(\dfrac{29}{15}\)

Let \(h(x) = \sqrt{f^2 + g^2}\). Then \(h'(x) = \dfrac{f(x)f'(x) + g(x)g'(x)}{\sqrt{f^2 + g^2}}\).

At \(x = 1\): \(\sqrt{9 + 16} = 5\), \(f f' = 3\left(-\frac13\right) = -1\), \(g g' = (-4)\left(-\frac83\right) = \frac{32}{3}\).

\(h'(1) = \dfrac{-1 + \frac{32}{3}}{5} = \dfrac{29/3}{5} = \dfrac{29}{15}\).

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

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Let \(f(1) = 3\), \(f'(1) = -\dfrac{1}{3}\), \(g(1) = -4\) and \(g'(1) = -\dfrac{8}{3}\). The derivative of \(\sqrt{[f(x)]^2 + [g(x)]^2}\) w.r.t. \(x\) at \(x = 1\) is _____.

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