First slide

Derivatives

Question

$If f (x) = \cos \cos 2 xcos 4 xcos 8 xcos 16 x then f^{1} (π / 4) equals$

Moderate

A

$\sqrt{2}$
B

sin 90
C

$\frac{1}{\sqrt{2}}$
D

$\cos e c \frac{π}{4}$

Solution

$\begin{array}{l} Cos A Cos 2 A Cos 4 A \dots \dots Cos 2^{n} A = \frac{\sin^{n + 1} x}{2^{n + 1} \sin x} \\ f (x) = \frac{\sin 2^{5} x}{2^{5} \sin x} = \frac{\sin 32 x}{32 \sin x} \\ f^{'} (x) = \frac{32 \sin x \cos 32 x - \cos x \sin 32 x}{32 \sin^{2} x} \\ f^{'} (\frac{π}{4}) = \frac{32 \sin \frac{π}{4} \cos 8 π - 0}{32 (\frac{1}{2})} \\ = 2 \sin \frac{π}{4} = \sqrt{2} = cosec \frac{π}{4} \end{array}$

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