Let $f:\mathbb R^2\to\mathbb R$ $$f(x,y)=xy+\frac{50}{x}+\frac{20}{y}.$$ Compute $\frac{\partial f}{\partial x},\;\frac{\partial f}{\partial y}$ and find local extrema of the function $f$.
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The partial derivative with respect to $x$ is calculated by thinking that $y$ is a constant. Therefore you have $$ \frac{{\partial f}} {{\partial x}} = y - \frac{{50}} {{x^2 }} $$ The partial derivative with respect to $y$ is calculated by thinking that $x$ is a constant. Therefore you have $$ \frac{{\partial f}} {{\partial y}} = x - \frac{{20}} {{y^2 }} $$