ABSTRACT
Let us suppose that x and y depend only on a single real variable t with
xX (t) and yY (t),
where X (t) and Y (t) are differentiable functions of t . Then in terms of
differentials
dxX ?(t)dt and dyY ?(t)dt,
so substituting into the expression for dw gives
dw @f
@x X ?(t)dt
@f
@y Y ?(t)dt:
Dividing by dt and using the fact that the quotient of the differentials dw
and dt is, by virtue of the definition of differentials, the derivative dw /dt , we
arrive at the result
dw
dt
@f
@x X ?(t)
@f
@y Y ?(t):
However,
X ?(t) dx
dt and Y ?(t)
dy
dt ,
so we find that
dw
dt
@f
@x
dx
dt
@f
@y
dy
dt :
that the function f (X (t), Y (t)) is a differentiable function with respect to t .
The chain rule also shows how the derivative dw /dt may be determined with-
out the need to differentiate w/f(X (t ), Y (t)) directly with respect to t . For obvious reasons, the derivative dw /dt computed from w/f(X (t), Y (t )) when both x and y are functions of t is called the total derivative.
