Question

Найти частные производные функций

Answer 1

$\displaystyle z=\ln (x+\sqrt{x^2+y^2})\\\\\\\frac{dz}{dx}=\frac{(x+\sqrt{x^2+y^2})'}{x+\sqrt{x^2+y^2}}=\frac{1+\frac{(x^2+y^2)'}{2\sqrt{x^2+y^2})}}{x+\sqrt{x^2+y^2}}=\boxed{\frac{1+\frac{x}{\sqrt{x^2+y^2})}}{x+\sqrt{x^2+y^2}}}\\\\\\\frac{dz}{dy}=\frac{(x+\sqrt{x^2+y^2})'}{x+\sqrt{x^2+y^2}}=\frac{\frac{(x^2+y^2)'}{2\sqrt{x^2+y^2})}}{x+\sqrt{x^2+y^2}}=\frac{\frac{2y}{2\sqrt{x^2+y^2})}}{x+\sqrt{x^2+y^2}}=\\\\\\=\frac{y}{(x+\sqrt{x^2+y^2})*\sqrt{x^2+y^2}}=\boxed{\frac{y}{x\sqrt{x^2+y^2}+x^2+y^2}}$