First slide

Derivatives

Question

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

Moderate

A

$2^{20} (\cos 2 x - 2^{20} \cos 3 x)$
B

$2^{20} (\cos 2 x + 2^{20} \cos 4 x)$
C

$2^{20} (\sin 2 x + 2^{20} \sin 4 x)$
D

$2^{20} (\sin 2 x - 2^{20} \sin 4 x)$

Solution

$y = 2 \cos x \cos 3 x = \cos 4 x + \cos 2 x$
$\begin{array}{l} \frac{d^{4} y}{d x^{4}} = 4^{4} \cos 4 x + 2^{4} \cos 2 x \\ \frac{d^{4 n} y}{d x^{4 n}} = 4^{4 n} \cos 4 x + 2^{4 n} \cos 2 x \\ \frac{d^{20} y}{d x^{20}} = 4^{20} \cos 4 x + 2^{20} \cos 2 x \end{array}$

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