Question

Помогите решить пожалуйста.
Найти dy/dx d^2y/dx^2 при значениях y=e^sqrt(x)

Answer 1

$displaystyle y=e^{sqrt{x}}\\frac{dy}{dx}=e^{sqrt{x}}*(sqrt{x})'=boxed{frac{e^{sqrt{x}}}{2sqrt{x}}}\\\frac{d^2y}{dx^2}=bigg(frac{e^{sqrt{x}}}{2sqrt{x}}bigg)'=frac{(e^{sqrt{x}})'*2sqrt{x}-(2sqrt{x})'e^{sqrt{x}}}{(2sqrt{x})^2}=\\\=frac{frac{e^{sqrt{x}}}{2sqrt{x}}*2sqrt{x}-frac{e^{sqrt{x}}}{sqrt{x}}}{4x}=e^{sqrt{x}}bigg(frac{1-frac{1}{sqrt{x}}}{4x}bigg)=boxed{e^{sqrt{x}}bigg(frac{sqrt{x}-1}{4xsqrt{x}}bigg)}$