Question

If $f ( x ) = ( \cos x + i \sin x ) ( \cos 3 x + i \sin 3 x ) \ldots$

$(\cos(2n - 1)x + i\sin(2n - 1)x),$ then $f^{'^{'}}$ is equal to

• A. $n^{2}f(x)$
• B. $- n^{4}f(x)$
• C. $- n^{2}f(x)$
• D. $n^{4}f(x)$

Answer 1

Answer: (B)

$- n^{4}f(x)$

Answer 2

We have,

}{+ .... + (2n - 1)x) = \cos n^{2}x + i\sin n^{2}x}$$ ⇒ $f^{'}(x) = - n^{2}(\sin n^{2}x) + n^{2}(i\cos n^{2}x)$ ⇒ $f^{'^{'4}{\cos n^{2}}^{4}\sin n^{2}}$ ⇒ $f^{'^{'4}\cos n^{2}\sin n^{2}}$ ⇒ $f^{'^{'4}}$