Determine the derivatives of the function 𝑓(𝑥,𝑦)=2𝑥𝑦−5𝑥3𝑦2+5𝑥+7𝑦 and obtain the values at point (2,3)



Answer to a math question Determine the derivatives of the function 𝑓(𝑥,𝑦)=2𝑥𝑦−5𝑥3𝑦2+5𝑥+7𝑦 and obtain the values at point (2,3)

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Para determinar as derivadas parciais da função 𝑓(𝑥,𝑦)=2𝑥𝑦−5𝑥^3𝑦^2+5𝑥+7𝑦 em relação a 𝑥 e 𝑦, podemos calcular as derivadas parciais de cada termo em relação a cada variável e então somá-las.

1. Derivada parcial em relação a 𝑥:
\frac{{\partial f}}{{\partial x}} = \frac{{\partial (2xy)}}{{\partial x}} - \frac{{\partial (5x^3y^2)}}{{\partial x}} + \frac{{\partial (5x)}}{{\partial x}} + \frac{{\partial (7y)}}{{\partial x}}
\frac{{\partial f}}{{\partial x}} = 2y - 15x^2y^2 + 5

2. Derivada parcial em relação a 𝑦:
\frac{{\partial f}}{{\partial y}} = \frac{{\partial (2xy)}}{{\partial y}} - \frac{{\partial (5x^3y^2)}}{{\partial y}} + \frac{{\partial (5x)}}{{\partial y}} + \frac{{\partial (7y)}}{{\partial y}}
\frac{{\partial f}}{{\partial y}} = 2x - 10x^3y + 7

Agora, para encontrar os valores no ponto (2,3), substituímos 𝑥=2 e 𝑦=3:

\frac{{\partial f}}{{\partial x}}\Big|_{(2,3)}=2(3)-15(2)^2(3)^2+5=-529

\frac{{\partial f}}{{\partial y}}\Big|_{(2,3)}=2(2)-10(2)^3(3)+7=-229

Portanto, as derivadas parciais da função 𝑓(𝑥,𝑦) em relação a 𝑥 e 𝑦 no ponto (2,3) são -169 e -469, respectivamente.

\textbf{Resposta:}\frac{{\partial f}}{{\partial x}}\Big|_{(2,3)}=-529\quad\text{e}\quad\frac{{\partial f}}{{\partial y}}\Big|_{(2,3)}=-229

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