use-the-first-principle-y-ln-cos-x-

Question Number 43854 by peter frank last updated on 16/Sep/18

u s e t h e f i r s t p r i n c i p l e y = \ln \sqrt{\cos x}

Answered by tanmay.chaudhury50@gmail.com last updated on 16/Sep/18

y + △ y = l n \sqrt{c o s (x + △ x)}

△ y = \frac{1}{2} {l n (c o s (x + △ x)} - \frac{1}{2} {l n c o s x}

△ y = \frac{1}{2} l n {\frac{c o s (x + △ x)}{c o s x}}

\frac{d y}{d x} = \lim_{△ x \to 0} \frac{△ y}{△ x}

= \frac{1}{2} \lim_{△ x \to 0} \frac{1}{2} \times \frac{l n {1 + \frac{c o s (x + △ x)}{c o s x} - 1}}{{\frac{c o s (x + △ x)}{c o s x} - 1}} \times \frac{c o s (x + △ x) - c o s x}{c o s x \times △ x}

l e t t = \frac{c o s (x + △ x)}{c o s x} - 1

w h e n △ x \to 0 t \to 0

s o

\frac{1}{2} \lim_{t \to 0} \times \frac{l n (1 + t)}{t} \times \lim_{△ x \to 0} \frac{2 s i n (x + \frac{△ x}{2}) s i n ((- \frac{△ x}{2})}{c o s x \times (\frac{- △ x}{2}) \times \frac{- 2}{}}

\frac{1}{2} \times 1 \times \frac{2 s i n x \times 1}{- 2 c o s x} = \frac{1}{2} \times - t a n x

r e c h e c k

\frac{d {\frac{1}{2} l n c o s x}}{d x} = \frac{1}{2} \times - t a n x