Намиране на производна

[Гост]

Производната на arctg(1+[tex]sin^{2 }[/tex]x)

[KOPMOPAH]

... e $\dfrac{2\,\cos\left(x\right)\,\sin\left(x\right)}{\sin^{4}\left(x\right)+2\,\sin^{2}\left(x\right)+2}=\cdots$

[ammornil]

Производната на arctg(1+[tex]sin^{2 }[/tex]x)

[tex](x)'=1 \\ u(x)=\sin{x} \rightarrow u'(x)=\cos{x} \\ v(x)=[u^{2}(x)] \rightarrow v'(x)=2\cdot{u(x)}=2\cdot\sin{x} \\ w(x)=\arctg{(1+v(x))} \rightarrow w'(x)=\frac{1}{1+(1+v(x))^{2}}=\frac{1}{1+(1+\sin^{2}{x})^{2}}=\frac{1}{\sin^{4}{x}+2\cdot{\sin^{2}{x}}+2}[/tex]
[tex](\arctg(1+\sin^{2}{x}))'=w'(x)\cdot{v'(x)}\cdot{u'(x)}\cdot{(x)'}=\frac{2\cdot{\sin{x}}\cdot{\cos{x}}}{\sin^{4}{x}+2\cdot{\sin^{2}{x}}+2}=\cdots[/tex]