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Differentiation — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDifferentiation3 Marks
Question
Differentiate the following w.r.t. x :$y=\frac{x^2 \sin x}{x+\cos x}$
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Answer

$y=\frac{x^2 \sin x}{x+\cos x}$
Differentiating w.r.t. $x$, we get
$\frac{ d y}{ d x}=\frac{ d }{ d x}\left(\frac{x^2 \sin x}{x+\cos x}\right)$
$=\frac{(x+\cos x) \frac{ d }{ d x}\left(x^2 \sin x\right)-x^2 \sin x \frac{ d }{ d x}(x+\cos x)}{(x+\cos x)^2}$
$=\frac{(x+\cos x)\left(x^2 \frac{ d }{ d x} \sin x+\sin x \frac{ d }{ d x} x^2\right)-x^2 \sin x\left(\frac{ d }{ d x} x+\frac{ d }{ d x} \cos x\right)}{(x+\cos x)^2}$
$=\frac{(x+\cos x)\left[x^2 \cos x+\sin x(2 x)\right]-x^2 \sin x(1-\sin x)}{(x+\cos x)^2}$
$=\frac{x^3 \cos x+2 x^2 \sin x+x^2 \cos ^2 x+2 x \sin x \cos x-x^2 \sin x+x^2 \sin ^2 x}{(x+\cos x)^2}$
$=\frac{x^3 \cos x+x^2 \sin x+x^2 \cos 2 x+x^2 \sin ^2 x+2 x \sin x \cos x}{(x+\cos x)^2}$
$=\frac{x^3 \cos x+x^2 \sin x+x^2\left(\sin { }^2 x+\cos ^2 x\right)+x \sin 2 x}{(x+\cos x)^2}$
$=\frac{x^2+x^2 \sin x+x^3 \cos x+x \sin 2 x}{(x+\cos x)^2}$
$=\frac{x^2(1+\sin x+x \cos x)+x \sin 2 x}{(x+\cos x)^2}$

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