Question

$\frac{d^{20}}{d{x}^{20}}\left(2\cos x\cos 3x\right)=$

(a) 2²⁰ (cos 2 x − 2²⁰ cos 4 x)
(b) 2²⁰ (cos 2 x + 2²⁰ cos 4 x)
(c) 2²⁰ (sin 2 x + 2²⁰ sin 4 x)
(d) 2²⁰ (sin 2 x − 2²⁰ sin 4 x)

Answer 1

(b) 2²⁰(cos2x + 2²⁰cos4x)

Here,
$$\begin{gathered} y=2\cos x\cos 3x=\cos \left(3x-x\right)+\cos \left(3x+x\right) \\ =\cos 2x+\cos 4x \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=-2\sin 2x-4\sin 4x=-2\left(\sin 2x+2\sin 4x\right) \\ \Rightarrow \frac{{\mathit{d}}^{2}y}{\mathit{d}{x}^2}=-4\cos 2x-16\cos 4x=-2^2\left(\cos 2x+2^2\cos 4x\right) \\ \Rightarrow \frac{{\mathit{d}}^{3}y}{\mathit{d}{x}^3}=2^3\left(\sin 2x+2^3\sin 4x\right) \\ \Rightarrow \frac{{\mathit{d}}^{4}y}{\mathit{d}{x}^4}=2^3\left(2\cos 2x+4\times 2^3\cos 4x\right)=2^4\left(\cos 2x+2^4\cos 4x\right) \\ \therefore \frac{{\mathit{d}}^{20}\left(\cos 2x+\cos 4x\right)}{\mathit{d}{x}^{20}}=2^{20}\left(\cos 2x+2^{20}\cos 4x\right) \end{gathered}$$