Question

Find the derivative of f(X)= cos(X)- sin(X) at X= 2π\3

Answer 1

$Given \:f(x) = cos x -sin x$

$f'(x) = \frac{d}{dx} (cos x - sin x )\\= \frac{d(cos x)}{dx} - \frac{d(sin x)}{dx} \\= - sinx - cos x \: --(1)$

$If \: x = \frac{2\pi }{3} \:then$

$\implies f'\Big(\frac{2\pi}{3}\Big) = - sin \Big(\frac{2\pi }{3}\Big) - cos \Big(\frac{2\pi }{3}\Big)\\= - sin \Big(\pi - \frac{\pi }{3}\Big) - cos \Big(\pi - \frac{\pi }{3}\Big)\\= - sin \Big(\frac{\pi }{3}\Big) + cos \Big(\frac{\pi }{3}\Big)\\= - \frac{\sqrt{3}}{2} + \frac{1}{2} \\= \frac{-\sqrt{3} + 1}{2}$

Therefore.,

$\red { f'\Big(\frac{2\pi}{3}\Big)} \green { = \frac{-\sqrt{3} + 1}{2}}$

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