Question

help differentiation I'll mark you the brainless please help I'm in a test rn​

Answer 1

$\sqrt{ \frac{1 - \cos x }{1 + \cos x } } + \tan ^{3} 4x$

$\implies \sqrt{ \frac{2 \sin ^{2} \frac{x}{2} }{2 \cos ^{2} \frac{x}{2} } } + \tan ^{3} 4x$

$\implies \sqrt{ (\frac{\sin \frac{x}{2} }{\cos \frac{x}{2} } ) ^{2} } + \tan ^{3} 4x$

$\implies \sqrt{ (\tan \frac{x}{2} ) ^{2} } + \tan ^{3} 4x$

$\implies \tan \frac{x}{2} + \tan ^{3} 4x$

$\implies \frac{d}{dx} (\tan \frac{x}{2} + \tan ^{3} 4x)$

$\implies \sec ^{2} \frac{x}{2} \cdot \frac{1}{2} + 3\tan ^{2} 4x \: \cdot \sec ^{2} 4x \: \cdot4$

$\orange{\therefore \frac{1}{2} \sec ^{2} \frac{x}{2} + 12\tan ^{2} 4x \: \sec ^{2} 4x }$