Задача 9 (Похідна складеної функції)

Обчислити похідну функції  y = arcsin\: e^{x}-\sqrt{1-e^{2x}}

 y' = (arcsin\: e^{x}-\sqrt{1-e^{2x}})'=

 =(arcsin\: e^{x})'-(\sqrt{1-e^{2x}})'=

 =\frac{1}{\sqrt{1-e^{2x}}}\cdot (e^{x})'-\frac{1}{2\sqrt{1-e^{2x}}}\cdot \left(1-e^{2x} \right)=

 =\frac{e^{x}}{\sqrt{1-e^{2x}}}-\frac{1}{2\sqrt{1-e^{2x}}}\cdot \left(-2e^{2x} \right)=

 =\frac{1}{\sqrt{1-e^{2x}}}\left(e^{x}+e^{2x} \right)=\frac{e^{x}+e^{2x}}{\sqrt{1-e^{2x}}}

2 Responses to “Задача 9 (Похідна складеної функції)”

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