Cot u(x)'in Türevi

Cot u(x)'in Türevi Nedir ?

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

$f(x)=cotu(x)\Rightarrow f^{\prime }(x)=-u^{\prime }(x).(1+co{t}^{2}u(x))$

$\frac{d(cotu(x))}{dx}=\frac{d}{dx}(cotu(x))=-u^{\prime }(x).(1+co{t}^{2}u(x))$

Cot u(x)'in Türevinin İspatı

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

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

$\boxed{cotx=\frac{cosx}{sinx}}$

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

$(cotu(x){)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{sinu(x).cosu(x+h)-sinu(x+h).cosu(x)}{sinu(x).sinu(x+h)}}{h}$

$(cotu(x){)}^{\prime }=\underset{h\to 0}{\lim }\frac{sinu(x).cosu(x+h)-sinu(x+h).cosu(x)}{h.sinu(x).sinu(x+h)}$

$\boxed{sinp.cosq-sinq.cosp=sin(p-q)}$

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

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

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

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

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

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

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

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

$(cotu(x){)}^{\prime }=-\underset{h\to 0}{\lim }\frac{u(x+h)-u(x)}{h}.\underset{h\to 0}{\lim }\frac{sinh}{h}.\underset{h\to 0}{\lim }\frac{1}{sinu(x).sinu(x+h)}$

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

$(cotu(x){)}^{\prime }=-u^{\prime }(x).1.\frac{1}{sinu(x).sinu(x+0)}$

$(cotu(x){)}^{\prime }=-u^{\prime }(x).\frac{1}{sinu(x).sinu(x)}$

$(cotu(x){)}^{\prime }=-\frac{u^{\prime }(x)}{si{n}^{2}u(x)}$

$(cotu(x){)}^{\prime }=-u^{\prime }(x).\frac{1}{si{n}^{2}u(x)}$

$(cotu(x){)}^{\prime }=-u(x).(\frac{1}{sinu(x)}{)}^2$

$\boxed{\frac{1}{sinx}=cscx}$

$(cotu(x){)}^{\prime }=-u(x).cs{c}^{2}u(x)$

$(cotu(x){)}^{\prime }=-u^{\prime }(x).\frac{1}{si{n}^{2}u(x)}$

$\boxed{1=si{n}^{2}x+co{s}^{2}x}$

$(cotu(x){)}^{\prime }=-u^{\prime }(x).(\frac{si{n}^{2}u(x)+co{s}^{2}u(x)}{si{n}^{2}u(x)})$

$(cotu(x){)}^{\prime }=-u^{\prime }(x).(\frac{\cancel{si{n}^{2}u(x)}}{\cancel{si{n}^{2}u(x)}}+\frac{co{s}^{2}u(x)}{si{n}^{2}u(x)})$

$(cotu(x){)}^{\prime }=-u^{\prime }(x).(1+(\frac{cosu(x)}{sinu(x)}{)}^2)$

$(cotu(x){)}^{\prime }=-u^{\prime }(x).(1+co{t}^{2}u(x))$