If f(x)=cos⁡x⋅cos⁡2x⋅cos⁡4x⋅cos⁡8x⋅cos⁡16x then the value of f′π4 is

A
1
B
$\sqrt{2}$
C
$\frac{1}{\sqrt{2}}$
D
0

Solution:

We know that,

$\begin{array}{l} \cos Acos 2 Acos 2^{2} A \dots \cos 2^{n - 1} A \\ = \frac{\sin (2^{n} A)}{2^{n} \sin A} \\ ∴ \cos xcos 2 xcos 4 xcos 8 xcos 16 x = \frac{\sin 32 x}{32 \sin x} \\ \Rightarrow f (x) = \frac{1}{32} \cdot \frac{\sin 32 x}{\sin x} \end{array}$

$\begin{array}{l} ∴ f^{'} (x) = \frac{1}{32} \times \frac{\sin x (32 \cos 32 x) - \sin 32 xcos x}{\sin^{2} x} \\ \Rightarrow f^{'} (\frac{π}{4}) = \frac{1}{\sin \frac{π}{4}} = \sqrt{2} \end{array}$

If $f (x) = \cos x \cdot \cos 2 x \cdot \cos 4 x \cdot \cos 8 x \cdot \cos 16 x$ then the value of $f^{'} (\frac{π}{4})$ is

Solution:

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