d^ 20 dx^ 20 (2 3x)= - Vidyadip

MCQ

$\frac{\text{d}^{20}}{\text{dx}^{20}}(2\cos\text{x}\cos3\text{x})=$

A
$2^{20}(\cos2\text{x}-2^{20}\cos4\text{x})$
✓
$2^{20}(\cos2\text{x}+2^{20}\cos4\text{x})$
C
$2^{20}(\sin2\text{x}+2^{20}\sin4^\text{x})$
D
$2^{20}(\sin2\text{x}-2^{20}\sin4^\text{x})$

✓

Answer

Correct option: B.

$2^{20}(\cos2\text{x}+2^{20}\cos4\text{x})$

$\text{y}=2\cos\text{x}\cos3\text{x}=\cos(3\text{x}-\text{x})+\cos(3\text{x}+\text{x})=\cos2\text{x}+\cos4\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-2\sin2\text{x}-4\sin4\text{x}=-2(\sin2\text{x}+2\sin4\text{x})$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=-4\cos2\text{x}-16\cos4\text{x}=-2^2(\cos2\text{x}+2^2\cos4\text{x})$
$\Rightarrow\frac{\text{d}^3\text{y}}{\text{dx}^3}=2^3(\sin2\text{x}+2^3\sin4\text{x})$
$\Rightarrow\frac{\text{d}^4\text{y}}{\text{dx}^4}=2^3(2\cos2\text{x}+4\times2^3\cos4\text{x})=2^4(\cos2\text{x}+2^4\cos4\text{x})$
$\therefore\frac{\text{d}^{20}(\cos2\text{x}+\cos4\text{x})}{\text{dx}^{20}}=2^{20}(\cos2\text{x}+2^{20}\cos4\text{x})$

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