The function y = y(x) defines a curve over the x-axis.
The derivative dy/dx is defined by
Graphically, dy/dx is the slope of the tangent line to the curve y(x). Equivalently, it is the coefficient in the linear approximation to y(x).
If y depends on x implicitly -- that is, if y = y(u) where u = u(x) -- the chain rule states that
The function z = z(x,y) defines a surface over the xy-plane.
On the surface, the point 
has neighbors in all directions. (By 
, we mean the
vector 
.) Particularly, consider the line 
in the xy-plane. This line determines an
image z(l), which is a curve along the surface z(x,y)
passing through p.
Note that the curve z(l) lies directly above the line l;
that is, it lies in the lz-plane. We can thus define the directional
derivative 
at p as follows: is the slope of the tangent line to the curve
z(l), where 
. This curve is simply the
image of the line l in the xy-plane, which starts at the pre-image
of p and points in the 
direction.
In particular, the directional derivative with respect to 
at
is simply the slope of the tangent
line to the image 
. We call this directional derivative of
z with respect to the axis the partial derivative
, which we also denote as 
. Since
it is just the slope of the curve , in an xz-plane with a
fixed value of 
, we readily see how to compute it. The value of
is given by the ordinary derivative dz/dx,
with the orthogonal variable y treated as a constant.
Sometimes we write this partial derivative as 
to indicate that y is an orthogonal
variable being held constant. Usually, though, the orthogonal
variables that are to be kept constant are obvious from context, so we
suppress the additional notation. Just remember that it's
dangerous to mix and match coordinate systems -- mixing independent
and dependent variables - when taking partial derivatives. ONLY the
orthogonal variables are to be held constant.
The chain rule takes 3 forms in multivariable calculus. The first is fundamental; the second two are obtained by limits of the first.
, the chain rule states that
, the chain rule states that
, (i.e.
the 
are functions of multiple variables 
), the chain rule
states that
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