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

Andrea
4.5
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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