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For a function of two variables $f(x,y)$, the first-order partial derivatives $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ are found by differentiating with respect to one variable while holding the other variable constant. Below I demonstrate the process on several representative functions and show the complete manual differentiation steps. 1. $f(x,y)=x^{3}y^{2}+\sin(xy)-e^{x}\cos y.$ Differentiate with respect to $x$ (treat $y$ as constant): $$ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}\big(x^{3}y^{2}\big)+\frac{\partial}{\partial x}\big(\sin(xy)\big)-\frac{\partial}{\partial x}\big(e^{x}\cos y\big). $$ For the first term: $y^{2}$ is constant so $\frac{\partial}{\partial x}(x^{3}y^{2})=3x^{2}y^{2}.$ For the second term: use chain rule on _____ ______ _______ ________ _________ _____ ______ ___ ______ ________ ________.
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