Sec x'in Türevi (Sekantın Türevi)

Sec x'in Türevi Nedir ?

Sec x'in türevi, tan x.sec x'dir.

$\frac{d}{dx}(\sec x)=\tan x.\sec x$

Sec x'in Türevinin İspatı

1. Yol

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\sec (x+h)-\sec x}{h}$

$\boxed{\sec x=\frac{1}{\cos x}}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{1}{\cos (x+h)}-\frac{1}{\cos x}}{h}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{\cos x-\cos (x+h)}{\cos x.\cos (x+h)}}{h}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\cos x-\cos (x+h)}{h.\cos x.\cos (x+h)}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{-[\cos (x+h)-\cos x]}{h.\cos x.\cos (x+h)}$

$\boxed{\cos p-\cos q=-2.\sin (\frac{p-q}{2}).\sin (\frac{p+q}{2})}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{-[-2.\sin (\frac{\cancel{x}+h-\cancel{x}}{2}).\sin (\frac{x+h+x}{2})]}{h.\cos x.\cos (x+h)}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{-[-2.\sin (\frac{h}{2}).\sin (\frac{2x+h}{2})]}{h.\cos x.\cos (x+h)}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{2.\sin (\frac{h}{2}).\sin (\frac{2x+h}{2})}{h.\cos x.\cos (x+h)}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{2.\sin (\frac{h}{2}).\sin (x+\frac{h}{2})}{h.\cos x.\cos (x+h)}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\sin (\frac{h}{2}).\sin (x+\frac{h}{2})}{\frac{h}{2}.\cos x.\cos (x+h)}$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }[\frac{\sin (\frac{h}{2})}{\frac{h}{2}}.\frac{\sin (x+\frac{h}{2})}{\cos x.\cos (x+h)}]$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\sin (\frac{h}{2})}{\frac{h}{2}}.\underset{h\to 0}{\lim }\frac{\sin (x+\frac{h}{2})}{\cos x.\cos (x+h)}$

$h=\frac{h}{2}$$(h\to 0)$

$(\sec x{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\sin h}{h}.\underset{h\to 0}{\lim }\frac{\sin (x+h)}{\cos x.\cos (x+h)}$

$\boxed{\underset{t\to 0}{\lim }\frac{\sin t}{t}=1}$

$(\sec x{)}^{\prime }=1.\frac{\sin (x+0)}{\cos x.\cos (x+0)}$

$(\sec x{)}^{\prime }=\frac{\sin (x+0)}{\cos x.\cos (x+0)}$

$(\sec x{)}^{\prime }=\frac{\sin x}{\cos x.\cos x}$

$(\sec x{)}^{\prime }=\frac{\sin x}{\cos x}.\frac{1}{\cos x}$

$\boxed{\tan x=\frac{\sin x}{\cos x}}$

$(\sec x{)}^{\prime }=\tan x.\sec x$

2. Yol

$\boxed{\sec x=\frac{1}{\cos x}}$

$(\sec x{)}^{\prime }=(\frac{1}{\cos x}{)}^{\prime }$

$\boxed{f(x)=\frac{u(x)}{v(x)}\Rightarrow f^{\prime }(x)=\frac{u^{\prime }(x).v(x)-v^{\prime }(x).u(x)}{[v(x){]}^2}}$

$(\sec x{)}^{\prime }=\frac{(1{)}^{\prime }.\cos x-(\cos x{)}^{\prime }.1}{{\cos }^{2}x}$

$\boxed{(\cos x{)}^{\prime }=-\sin x}$

$(\sec x{)}^{\prime }=\frac{0.\cos x-(-\sin x).1}{{\cos }^{2}x}$

$(\sec x{)}^{\prime }=\frac{0-(-\sin x)}{{\cos }^{2}x}$

$(\sec x{)}^{\prime }=\frac{0+\sin x}{{\cos }^{2}x}$

$(\sec x{)}^{\prime }=\frac{\sin x}{{\cos }^{2}x}$

$(\sec x{)}^{\prime }=\frac{\sin x}{\cos x.\cos x}$

$(\sec x{)}^{\prime }=\frac{\sin x}{\cos x}.\frac{1}{\cos x}$

$\boxed{\tan x=\frac{\sin x}{\cos x}}$

$(\sec x{)}^{\prime }=\tan x.\sec x$