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Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation2 Marks
Question
Differentiate the following w. r. t. x.$\cos ^{-1}\left(2 x^2-x\right)$
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Answer

Let $y=\cos ^{-1}\left(2 x^2-x\right)$
Hence $\cos y=2 x^2-x$
Differentiate w.r.t. $x$.
$
-\sin y \cdot \frac{d y}{d x}=4 x-1
$
$
\begin{aligned}
\frac{d y}{d x} & =\frac{1-4 x}{\sin y}=\frac{1-4 x}{\sqrt{1-\cos ^2 y}} \\
\therefore \frac{d y}{d x} & =\frac{1-4 x}{\sqrt{1-x^2(2 x-1)^2}} \quad \ldots \text { from (I) }
\end{aligned}
$
... from (I)
Alternate Method :
$
\begin{aligned}
& \text { If } y=\cos ^{-1}\left(2 x^2-x\right) \\
& \text { Differentiate w.r.t.x. } \\
& \begin{aligned}
\frac{d y}{d x} & =\frac{d}{d x}\left(\cos ^{-1}\left(2 x^2-x\right)\right) \\
& =\frac{-1}{\sqrt{1-\left(2 x^2-x\right)^2}} \cdot \frac{d}{d x}\left(2 x^2-x\right) \\
& =\frac{-1}{\sqrt{1-x^2(2 x-1)^2}} \cdot(4 x-1) \\
\therefore \quad \frac{d y}{d x} & =\frac{1-4 x}{\sqrt{1-x^2(2 x-1)^2}}
\end{aligned}
\end{aligned}
$

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