Lecture 8 starts a new unit in the 18.02 series and is about functions of several variables, the graph of a function of two variables, contour plots of functions of two variables and partial derivatives.
The image above shows the graph and the contour plot of $f(x,y)=1-(x^2+y^2)$ followed by the partial derivatives as calculated by Mathematica.
Partial derivatives are defined as follows:
$$\frac{\partial f}{\partial x}(x,y) = \displaystyle\lim_{\Delta x \to 0}\frac{f(x+\Delta x, y) - f(x,y)}{\Delta x}$$
$$\frac{\partial f}{\partial y}(x,y) = \displaystyle\lim_{\Delta y \to 0}\frac{f(x, y+\Delta y) - f(x,y)}{\Delta y}$$
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