9.3: Partial Derivatives.

Main Points
The partial derivative of f wrt x at (a,b) is the derivative of f with y constant. The reverse is also true (ie. wrt y with x constant.) This is computed using the formula: lim (h -> 0) (f(a + h, b) - f(a,b))/h. This is when y is fixed. When x is fixed the formula changes to lim (h -> 0) (f(a, b + h) - f(a,b))/h. Partial derivatives have a different symbol from the normal one. On a contour diagram, the partial derivative is taken as the rate of change of the value of the function on the contours. And as always the units of the partial derivative would help you to identify what exactly it is discussing.

Challenges
I kinda vaguely understood what the partial derivatives were in terms of making a two-variable equation a one-variable one by fixing one of the variables, but then Example 2 lost me. I have no idea how the values are computed or what actually they mean.

Reflections
Calculus and math for that matter can be used for a lot of things from the simple to the complex all in aid of better understanding it...But are these really used? With the airplane example, its pretty obvious that fixing x at 10 would cause the sales value to increase by $350 for each full priced ticket that is bought. Is that not like a "duh factor"? Are we assuming that figured out that increase was using partial derivatives?