Consider the function at the point (1/2, 1). We calculate the following limits
These partial derivatives are calculated in the same way as derivatives are calculated for functions of a single variable with the exception that we treat the other variable as a constant.
The following example shows this
f(x, y) = x cos(xy)Other Notations
fx = cos(xy) + x (sin(xy) y) = cos(xy) xy sin(xy)
fy = x (sin(xy) x) = x2 sin(xy)
Below is the graph of the function f(x, y) = x e( x2 y2) and following that are the graphs of fx and fy. Can you tell which is which?
There are higher order derivatives these have the notations
It should come as no surprise that if the partial derivatives are all continuous then fxy = fyx. Below are the graphs of fxx, fyy, and fxy for the function f(x, y) = x e( x2 y2)
fxx | fyy |
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fxy |
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