find the derivative of f(x)=root cosx by first prnciple - Maths -

Question

find the derivative of f(x)=root cosx by first
prnciple?????????

Ayush Jha · Accepted Answer

we have

$f (x) = \sqrt{\cos x}$

So,

$f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} = \lim_{h \to 0} \frac{\sqrt{\cos (x + h)} - \sqrt{\cos x}}{h} R a t i o n a l i s e - = \lim_{h \to 0} \frac{\cos (x + h) + \cos x}{h (\sqrt{\cos (x + h)} + \sqrt{\cos x})} = (\frac{1}{2 \sqrt{\cos x}}) \lim_{h \to 0} \frac{\cos x (\cosh - 1) - \sin x \sinh}{h} = (\frac{1}{2 \sqrt{\cos x}}) [\lim_{h \to 0} (\frac{\cos x (\cosh - 1)}{h}) - \lim_{h \to 0} (\frac{\sin x \sinh}{h})] = - \frac{\sin x}{2 \sqrt{\cos x}}$

find the derivative of f(x)=root cosx by first prnciple?????????