Question

Differentiate \(\sqrt{\frac{x+1}{3x-1}}\) with respect to x.

(a) \(\frac{-2}{(3x-1)\sqrt{(3x-1)(x+1)}}\)

(b) \(\frac{2}{(3x-1)\sqrt{(3x-1)(x+1)}}\)

(c) \(\frac{1}{(3x-1)\sqrt{(3x-1)(x+1)}}\)

(d) \(\frac{-2}{\sqrt{(3x-1)(x+1)}}\)

This question was posed to me in examination.

The question is from Logarithmic Differentiation in portion Continuity and Differentiability of Mathematics – Class 12

Answer 1

Correct choice is (a) \(\frac{-2}{(3x-1)\sqrt{(3x-1)(x+1)}}\)

The explanation: Consider y=\(\sqrt{\frac{x+1}{3x-1}}\)

Applying log to both sides, we get

log⁡y=log⁡\(\sqrt{\frac{x+1}{3x-1}}\)

log⁡y=\(\frac{1}{2} log⁡\left (\frac{x+1}{3x-1}\right )\)

log⁡y=\(\frac{1}{2}\) (log⁡(x+1)-log⁡(3x-1))

Differentiating with respect to x, we get

\(\frac{1}{y} \frac{dy}{dx}\)=\(\frac{1}{2}\left (\frac{d}{dx} (log⁡(x+1))-\frac{d}{dx} (log⁡(3x-1))\right )\)

\(\frac{1}{y} \frac{dy}{dx}\)=\(\frac{1}{2}\left (\frac{1}{x+1}-\frac{3}{3x-1}\right )\)

\(\frac{1}{y} \frac{dy}{dx}\)=\(\frac{1}{2}\left (\frac{3x-1-3x-3}{(x+1)(3x-1)})\right )\)

\(\frac{1}{y} \frac{dy}{dx}\)=\(\frac{1}{2}\left (\frac{-4}{(x+1)(3x-1)}\right )\)

\(\frac{dy}{dx}\)=\(\sqrt{\frac{x+1}{3x-1}} \left (\frac{-2}{(x+1)(3x-1)}\right )\)

\(\frac{dy}{dx}\)=\(\frac{-2}{(3x-1) \sqrt{(3x-1)(x+1)}}\)