Cot u(x)'in Türevi

Cot u(x)'in Türevi Nedir ?

Cot u(x)'in türevi, -u'(x).[1+cot² u(x)]'dir.

$\frac{d}{dx}[\cot u(x)]=-\frac{d}{dx}[u(x)].[1+{\cot }^{2}u(x)]$

Cot u(x)'in Türevinin İspatı

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

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

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

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

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

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

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

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

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

$\boxed{\sin (-x)=-\sin x}$

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

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

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

$[\cot u(x){]}^{\prime }=-\underset{h\to 0}{\lim }\{\frac{\sin [u(x+h)-u(x)]}{u(x+h)-u(x)}.\frac{u(x+h)-u(x)}{h}.\frac{1}{\sin u(x).\sin u(x+h)}\}$

$[\cot u(x){]}^{\prime }=-\underset{h\to 0}{\lim }\frac{\sin [u(x+h)-u(x)]}{u(x+h)-u(x)}.\underset{h\to 0}{\lim }\frac{u(x+h)-u(x)}{h}.\underset{h\to 0}{\lim }\frac{1}{\sin u(x).\sin u(x+h)}$

$h=u(x+h)-u(x)$$(h\to 0)$

$[\cot u(x){]}^{\prime }=-\underset{h\to 0}{\lim }\frac{\sin h}{h}.\underset{h\to 0}{\lim }\frac{u(x+h)-u(x)}{h}.\underset{h\to 0}{\lim }\frac{1}{\sin u(x).\sin u(x+h)}$

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

$[\cot u(x){]}^{\prime }=-1.u^{\prime }(x).\frac{1}{\sin u(x).\sin u(x+0)}$

$[\cot u(x){]}^{\prime }=-1.u^{\prime }(x).\frac{1}{\sin u(x).\sin u(x)}$

$[\cot u(x){]}^{\prime }=-1.u^{\prime }(x).\frac{1}{{\sin }^{2}u(x)}$

$[\cot u(x){]}^{\prime }=-u^{\prime }(x).\frac{1}{{\sin }^{2}u(x)}$

$\boxed{{\sin }^{2}x+{\cos }^{2}x=1}$

$[\cot u(x){]}^{\prime }=-u^{\prime }(x).[\frac{{\sin }^{2}u(x)+{\cos }^{2}u(x)}{{\sin }^{2}u(x)}]$

$[\cot u(x){]}^{\prime }=-u^{\prime }(x).[\frac{\cancel{{\sin }^{2}u(x)}}{\cancel{{\sin }^{2}u(x)}}+\frac{{\cos }^{2}u(x)}{{\sin }^{2}u(x)}]$

$[\cot u(x){]}^{\prime }=-u^{\prime }(x).\{1+[\frac{\cos u(x)}{\sin u(x)}{]}^2\}$

$[\cot u(x){]}^{\prime }=-u^{\prime }(x).[1+{\cot }^{2}u(x)]$

$Soru:$

$f(x)=\cot (4x-3)\Rightarrow f^{\prime }(x)=?$

$Cevap:$

$f(x)=\cot (4x-3)$

$f^{\prime }(x)=[\cot (4x-3){]}^{\prime }$

$\boxed{f(x)=\cot u(x)\Rightarrow f^{\prime }(x)=-u^{\prime }(x).[1+{\cot }^{2}u(x)]}$

$f^{\prime }(x)=-(4x-3{)}^{\prime }.[1+{\cot }^2(4x-3)]$

$f^{\prime }(x)=-(4-0).[1+{\cot }^2(4x-3)]$

$f^{\prime }(x)=-4.[1+{\cot }^2(4x-3)]$