Level Curves and Partial Derivatives

Level curves and partial derivatives. Partial derivatives: definitions

Partial derivatives are a notation used to express the derivative of a function with
respect to one of its variables, holding other variables constant. A function of several
variables may have partial derivatives with respect to each variable, but it does not
have a derivative in the usual sense.
δf/δx(x_0,y_0)=lim_(Δx->0) (f(x_0+Δx,y_0)-f(x_0,y_0))/Δx
partial
δf/δx(x_0,y_0)=lim_(Δy->0) (f(x_0,y_0+Δy)-f(x_0,y_0))/Δy
This symbol is a capital d with a curly line on the bottom.

It is not a straight line and it
is not d. a letter d. It is referred to as a "partial differential" or a "del" for short.
The partial derivative of a function f with respect to one of its variables, x , at the
point (x0,y0), is defined by the limit of this ratio as the change in x becomes
infinitesimally small.
So here we are actually not changing y at all. We are just changing x and looking at
the rate of change with respect to x. And we have the same with respect to y, partial f
partial y is the limit, so we should say at the point (x0, y0) is the limit as delta y turns as
to 0.
A partial derivative is a derivative of a function with respect to one of its variables,
but not the others.

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A function is differentiable if these partial derivatives exist.
So most functions are differentiable, and we will learn how to compute their partial
derivatives without having to use the usual methods.