Нужна помощь. найти полный дифференциал функции f(x,y,z)=x²/(y²x + z²).

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Нужна помощь. найти полный дифференциал функции f(x,y,z)=x²/(y²x + z²).

SandBuck 15.02.2026 14:20

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Лебедев Дмитрий Сергеевич · Accepted Answer

Для нахождения полного дифференциала функции трех переменных $f (x, y, z) f of open paren x comma y comma z close paren$ используется формула: $d f = \frac{f}{x} d x + \frac{f}{y} d y + \frac{f}{z} d z d f equals partial f over partial x end-fraction d x plus partial f over partial y end-fraction d y plus partial f over partial z end-fraction d z$ 1. Нахождение частных производных Функция имеет вид: $f (x, y, z) = \frac{x^{2}}{y^{2} x + z^{2}} f of open paren x comma y comma z close paren equals the fraction with numerator x squared and denominator y squared x plus z squared end-fraction$ . По переменной $x x$ : Используем правило дифференцирования дроби ${(\frac{u}{v})}^{'} = \frac{u^{'} v - u v^{'}}{v^{2}} open paren u over v end-fraction close paren prime equals the fraction with numerator u prime v minus u v prime and denominator v squared end-fraction$ . $\frac{f}{x} = \frac{(x^{2})_{x}^{'} \cdot (y^{2} x + z^{2}) - x^{2} \cdot (y^{2} x + z^{2})_{x}^{'}}{(y^{2} x + z^{2})^{2}} partial f over partial x end-fraction equals the fraction with numerator open paren x squared close paren sub x prime center dot open paren y squared x plus z squared close paren minus x squared center dot open paren y squared x plus z squared close paren sub x prime and denominator open paren y squared x plus z squared close paren squared end-fraction$ $\frac{f}{x} = \frac{2 x (y^{2} x + z^{2}) - x^{2} (y^{2})}{(y^{2} x + z^{2})^{2}} = \frac{2 x y^{2} x + 2 x z^{2} - x^{2} y^{2}}{(y^{2} x + z^{2})^{2}} = \frac{x^{2} y^{2} + 2 x z^{2}}{(y^{2} x + z^{2})^{2}} partial f over partial x end-fraction equals the fraction with numerator 2 x open paren y squared x plus z squared close paren minus x squared open paren y squared close paren and denominator open paren y squared x plus z squared close paren squared end-fraction equals the fraction with numerator 2 x y squared x plus 2 x z squared minus x squared y squared and denominator open paren y squared x plus z squared close paren squared end-fraction equals the fraction with numerator x squared y squared plus 2 x z squared and denominator open paren y squared x plus z squared close paren squared end-fraction$ По переменной $y y$ : Здесь $x x$ и $z z$ — константы. Представим функцию как $x^{2} \cdot (y^{2} x + z^{2})^{-1} x squared center dot open paren y squared x plus z squared close paren to the negative 1 power$ . $\frac{f}{y} = x^{2} \cdot (-1) (y^{2} x + z^{2})^{-2} \cdot (y^{2} x + z^{2})_{y}^{'} partial f over partial y end-fraction equals x squared center dot open paren negative 1 close paren open paren y squared x plus z squared close paren to the negative 2 power center dot open paren y squared x plus z squared close paren sub y prime$ $\frac{f}{y} = - \frac{x^{2}}{(y^{2} x + z^{2})^{2}} \cdot (2 y x) = - \frac{2 x^{3} y}{(y^{2} x + z^{2})^{2}} partial f over partial y end-fraction equals negative the fraction with numerator x squared and denominator open paren y squared x plus z squared close paren squared end-fraction center dot open paren 2 y x close paren equals negative the fraction with numerator 2 x cubed y and denominator open paren y squared x plus z squared close paren squared end-fraction$ По переменной $z z$ : Аналогично дифференцированию по $y y$ . $\frac{f}{z} = x^{2} \cdot (-1) (y^{2} x + z^{2})^{-2} \cdot (y^{2} x + z^{2})_{z}^{'} partial f over partial z end-fraction equals x squared center dot open paren negative 1 close paren open paren y squared x plus z squared close paren to the negative 2 power center dot open paren y squared x plus z squared close paren sub z prime$ $\frac{f}{z} = - \frac{x^{2}}{(y^{2} x + z^{2})^{2}} \cdot (2 z) = - \frac{2 x^{2} z}{(y^{2} x + z^{2})^{2}} partial f over partial z end-fraction equals negative the fraction with numerator x squared and denominator open paren y squared x plus z squared close paren squared end-fraction center dot open paren 2 z close paren equals negative the fraction with numerator 2 x squared z and denominator open paren y squared x plus z squared close paren squared end-fraction$ 2. Составление полного дифференциала Подставляем найденные производные в общую формулу: $d f = \frac{x^{2} y^{2} + 2 x z^{2}}{(y^{2} x + z^{2})^{2}} d x - \frac{2 x^{3} y}{(y^{2} x + z^{2})^{2}} d y - \frac{2 x^{2} z}{(y^{2} x + z^{2})^{2}} d z d f equals the fraction with numerator x squared y squared plus 2 x z squared and denominator open paren y squared x plus z squared close paren squared end-fraction d x minus the fraction with numerator 2 x cubed y and denominator open paren y squared x plus z squared close paren squared end-fraction d y minus the fraction with numerator 2 x squared z and denominator open paren y squared x plus z squared close paren squared end-fraction d z$ Для удобства можно вынести общий знаменатель за скобки: $d f = \frac{1}{(y^{2} x + z^{2})^{2}} ((x^{2} y^{2} + 2 x z^{2}) d x - 2 x^{3} y d y - 2 x^{2} z d z) d f equals the fraction with numerator 1 and denominator open paren y squared x plus z squared close paren squared end-fraction open paren open paren x squared y squared plus 2 x z squared close paren d x minus 2 x cubed y space d y minus 2 x squared z space d z close paren$ Ответ: $d f = \frac{x (x y^{2} + 2 z^{2}) d x - 2 x^{3} y d y - 2 x^{2} z d z}{(y^{2} x + z^{2})^{2}} d f equals the fraction with numerator x open paren x y squared plus 2 z squared close paren d x minus 2 x cubed y space d y minus 2 x squared z space d z and denominator open paren y squared x plus z squared close paren squared end-fraction$ Хотите, чтобы я вычислил значение этого дифференциала в конкретной точке при заданных приращениях?

Нужна помощь. найти полный дифференциал функции f(x,y,z)=x²/(y²x + z²).

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