Level Curves and Partial Derivatives. Computing Partial Derivatives

Level curves and partial derivatives. Computing partial derivatives

There is a notation used in physics and applied math that allows one to compute
these things. So how to compute?
How to compute?
to find δf/δx=f_x?
treat y as constant
x as variable
Ex: f(x, y)=x^3 y+y^2; δf/δx=3x^2 y+0
δf/δy=x^3 +2y
In Mathematics, the same symbol may be used for both a variable and one of its
derivatives. For example, f(x) is the derivative of f(x), where y is treated as a
constant and x is treated as a variable.
Furthermore, if we want to find partial derivatives with respect to y, we can simply
differentiate the equation with respect to x.
Let's consider the function f(x, y) = x3y + y2. Then we can take its partial derivatives
with respect to x and y. To find the derivative of a function that is raised to the third
power, you must take the derivative of each term and apply the multiplication rule.

The derivative of x cubed is 3 times x squared times the constant plus 0, because y
squared is a constant. If you instead consider a partial derivative, then the
expression is a constant times y. The derivative of y is 1, so that's just x cubed. The
derivative of y squared is 2 y.