Differentiate the following w.r.t. x: ^ -1 (1 4x^3-3x ),0<x<1 2 - Vidyadip

Question

Differentiate the following w.r.t. x:
$\sec^{-1}\Big(\frac{1}{4\text{x}^3-3\text{x}}\Big),0<\text{x}<\frac{1}{\sqrt{2}}$

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Answer

Let $\text{y}=\sec^{-1}\Big(\frac{1}{4\text{x}^3-3\text{x}}\Big)$
On putting $\text{x}=\cos\theta,$ we get
$\text{y}=\sec^{-1}\frac{1}{4\cos^3\theta-3\cos\theta}$
$=\sec^{-1}\frac{1}{\cos3\theta}$
$=\sec^{-1}(\sec3\theta)$
$=3\theta$
$=3\cos^{-1}\text{x}$ $\big[\because\theta=\cos^{-1}\text{x}\big]$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(3\cos^{-1}\text{x})$
$=\frac{-3}{\sqrt{1-\text{x}^2}}$

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