Предмет: Алгебра, автор: Deil2021

Найти частные производные по х, по у и по хх:
z=5^x+5^y+5+5^(x+y)+5^(5x+y-5)

Ответы

Автор ответа: Artem112

$z=5^x+5^y+5+5^{x+y}+5^{5x+y-5}$

Частная производная по выбранной переменной вычисляется в предположении, что все остальные переменные - константы.

$z'_x=(5^x+5^y+5+5^{x+y}+5^{5x+y-5})'_x$

$z'_x=5^x\ln5+0+0+5^{x+y}\ln5\cdot(x+y)'_x+5^{5x+y-5}\ln5\cdot(5x+y-5)'_x$

$z'_x=5^x\ln5+5^{x+y}\ln5\cdot1+5^{5x+y-5}\ln5\cdot5$

$z'_x=5^x\ln5+5^{x+y}\ln5+5^{5x+y-4}\ln5$

$\boxed{z'_x=(5^x+5^{x+y}+5^{5x+y-4})\ln5}$

$z'_y=(5^x+5^y+5+5^{x+y}+5^{5x+y-5})'_y$

$z'_y=0+5^y\ln5+0+5^{x+y}\ln5\cdot(x+y)'_y+5^{5x+y-5}\ln5\cdot(5x+y-5)'_y$

$z'_y=5^y\ln5+5^{x+y}\ln5\cdot1+5^{5x+y-5}\ln5\cdot1$

$z'_y=5^y\ln5+5^{x+y}\ln5+5^{5x+y-5}\ln5$

$\boxed{z'_y=(5^y+5^{x+y}+5^{5x+y-5})\ln5}$

$z''_{xx}=(z'_x)'_x$

$z''_{xx}=((5^x+5^{x+y}+5^{5x+y-4})\ln5)'_x$

$z''_{xx}=(5^x+5^{x+y}+5^{5x+y-4})'_x\ln5$

$z''_{xx}=(5^x\ln5+5^{x+y}\ln5\cdot(x+y)'_x+5^{5x+y-4}\ln5\cdot(5x+y-4)'_x)\ln5$

$z''_{xx}=(5^x\ln5+5^{x+y}\ln5\cdot1+5^{5x+y-4}\ln5\cdot5)\ln5$

$z''_{xx}=(5^x\ln5+5^{x+y}\ln5+5^{5x+y-3}\ln5)\ln5$

$\boxed{z''_{xx}=(5^x+5^{x+y}+5^{5x+y-3})\ln^25}$

Deil2021: спасибо

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Найти частные производные по х, по у и по хх: z=5^x+5^y+5+5^(x+y)+5^(5x+y-5)

Ответы

Найти частные производные по х, по у и по хх:
z=5^x+5^y+5+5^(x+y)+5^(5x+y-5)