Giải bài 49 trang 112 SGK giải tích nâng cao 12
Tính đạo hàm của các hàm số sau:
a) \(y=\left( x-1 \right){{e}^{2x}};\) b) \(y={{x}^{2}}\sqrt{{{e}^{4x}}+1};\)
c) \(y=\dfrac{1}{2}\left( {{e}^{x}}-{{e}^{-x}} \right);\) d) \(y=\dfrac{1}{2}\left( {{e}^{x}}+{{e}^{-x}} \right)\).
\(\begin{align} a)\,& y=\left( x-1 \right){{e}^{2x}} \\ y' &={{e}^{2x}}+2\left( x-1 \right){{e}^{2x}} \\ & =\left( 2x-1 \right){{e}^{2x}} \\ b)\,& y={{x}^{2}}\sqrt{{{e}^{4x}}+1} \\ y' &=2x\sqrt{{{e}^{4x}}+1}+{{x}^{2}}.\dfrac{4{{e}^{4x}}}{2\sqrt{{{e}^{4x}}+1}} \\ & =\dfrac{4x\left( {{e}^{4x}}+1 \right)+4{{x}^{2}}{{e}^{4x}}}{2\sqrt{{{e}^{4x}}+1}} \\ & =\dfrac{2x\left[ \left( x+1 \right){{e}^{4x}}+1 \right]}{\sqrt{{{e}^{4x}}+1}} \\ c)\,& y=\dfrac{1}{2}\left( {{e}^{x}}-{{e}^{-x}} \right) \\ y' &=\dfrac{1}{2}\left( {{e}^{x}}+{{e}^{-x}} \right) \\ d)\, & y=\dfrac{1}{2}\left( {{e}^{x}}+{{e}^{-x}} \right) \\ y' &=\dfrac{1}{2}\left( {{e}^{x}}-{{e}^{-x}} \right) \\ \end{align} \)