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.