The derivative of a function $g(x)$ of one variable is the rate of change of $g$ as $x$ changes. If we have a function $f(x,y)$ of two variables, remember the sixth pillar of calculus: One variable at a time! We can ask what happens when only $x$ changes, or what happens when only $y$ changes. These are called partial derivatives, and their definitions are $$ \frac{\partial f}{\partial x}(a,b) = \lim_{h \to 0} \frac{f(a+h,b)-f(a,b)}{h},$$ $$ \frac{\partial f}{\partial y}(a,b) = \lim_{h \to 0} \frac{f(a,b+h)-f(a,b)}{h}.$$

An example worked in the video is the function $f(x,y) = e^{xy^2}$. To take the derivative with respect to $x$, we treat $y$ as a constant, so $f(x,y) = e^{\hbox{constant times }x}$, where the "constant" is $y^2$. But we already know that the derivative of $e^{cx}$ is $c e^{x}$, so the partial derivative of $e^{xy^2}$ with respect to $x$ is $$f_x(x,y) = y^2 e^{xy^2}.$$ Similarly, we compute $f_y$ by treating $x$ as a constant. Then $f(x,y) = e^{\hbox{constant times }y^2}$, which we know how to take the derivative of with the help of the chain rule: $$f_y(x,y) = 2xy e^{xy^2}.$$