Sin u(x)'in Türevi

Sin u(x)'in Türevi Nedir ?

Sin u(x)'in türevi u'(x).cos u(x)'tir.

$f(x)=sinu(x)\Rightarrow f^{\prime }(x)=u^{\prime }(x).cosu(x)$

$\frac{d(sinu(x))}{dx}=\frac{d}{dx}(sinu(x))=u^{\prime }(x).cosu(x)$

Sin u(x)'in Türevinin İspatı

1. Yol

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

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

$\boxed{sinp-sinq=2.sin\frac{p-q}{2}.cos\frac{p+q}{2}}$

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

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

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

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

$\frac{u(x+h)-u(x)}{2}=h(h\to 0)$

$(sinu(x){)}^{\prime }=\underset{h\to 0}{\lim }\frac{u(x+h)-u(x)}{h}.\underset{h\to 0}{\lim }\frac{sinh}{h}.\underset{h\to 0}{\lim }cos\frac{u(x+h)+u(x)}{2}$

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

$(sinu(x){)}^{\prime }=u^{\prime }(x).1.cos\frac{u(x+0)+u(x)}{2}$

$(sinu(x){)}^{\prime }=u^{\prime }(x).cos\frac{u(x)+u(x)}{2}$

$(sinu(x){)}^{\prime }=u^{\prime }(x).cos\frac{\cancel{2}.u(x)}{\cancel{2}}$

$(sinu(x){)}^{\prime }=u^{\prime }(x).cosu(x)$

2. Yol

$\boxed{sinu(x)=u(x)-\frac{(u(x){)}^3}{3!}+\frac{(u(x){)}^5}{5!}-\frac{(u(x){)}^7}{7!}+\frac{(u(x){)}^9}{9!}-...}$

$\boxed{cosu(x)=1-\frac{(u(x){)}^2}{2!}+\frac{(u(x){)}^4}{4!}-\frac{(u(x){)}^6}{6!}+\frac{(u(x){)}^8}{8!}-...}$

$sinu(x)=u(x)-\frac{(u(x){)}^3}{3!}+\frac{(u(x){)}^5}{5!}-\frac{(u(x){)}^7}{7!}+\frac{(u(x){)}^9}{9!}-...$

$(sinu(x){)}^{\prime }=(u(x)-\frac{(u(x){)}^3}{3!}+\frac{(u(x){)}^5}{5!}-\frac{(u(x){)}^7}{7!}+\frac{(u(x){)}^9}{9!}-...{)}^{\prime }$

$(sinu(x){)}^{\prime }=u^{\prime }(x)-(\frac{(u(x){)}^3}{3!}{)}^{\prime }+(\frac{(u(x){)}^5}{5!}{)}^{\prime }-(\frac{(u(x){)}^7}{7!}{)}^{\prime }+(\frac{(u(x){)}^9}{9!}{)}^{\prime }-...$

$(sinu(x){)}^{\prime }=u^{\prime }(x)-\frac{3.(u(x){)}^2.u^{\prime }(x)}{3!}+\frac{5.(u(x){)}^4.u^{\prime }(x)}{5!}-\frac{7.(u(x){)}^6.u^{\prime }(x)}{7!}+\frac{9.(u(x){)}^8.u^{\prime }(x)}{9!}-...$

$(sinu(x){)}^{\prime }=u^{\prime }(x)-\frac{\cancel{3}.(u(x){)}^2.u^{\prime }(x)}{\cancel{3}.2!}+\frac{\cancel{5}.(u(x){)}^4.u^{\prime }(x)}{\cancel{5}.4!}-\frac{\cancel{7}.(u(x){)}^6.u^{\prime }(x)}{\cancel{7}.6!}+\frac{\cancel{9}.(u(x){)}^8.u^{\prime }(x)}{\cancel{9}.8!}-...$

$(sinu(x){)}^{\prime }=u^{\prime }(x).(1-\frac{(u(x){)}^2}{2!}+\frac{(u(x){)}^4}{4!}-\frac{(u(x){)}^6}{6!}+\frac{(u(x){)}^8}{8!}-...)$

$(sinu(x){)}^{\prime }=u^{\prime }(x).cosu(x)$