5.2 THE TOTAL DIFFERENTIAL AND TOTAL DERIVATIVE
Only three of the second derivatives are independent since the relation
∂^2 f
∂x∂y=∂^2 f
∂y∂x,is always obeyed, provided that the second partial derivatives are continuous
at the point in question. This relation often proves useful as a labour-saving
device when evaluating second partial derivatives. It can also be shown that for
a function ofnvariables,f(x 1 ,x 2 ,...,xn), under the same conditions,
∂^2 f
∂xi∂xj=∂^2 f
∂xj∂xi.Find the first and second partial derivatives of the functionf(x, y)=2x^3 y^2 +y^3.The first partial derivatives are
∂f
∂x=6x^2 y^2 ,∂f
∂y=4x^3 y+3y^2 ,and the second partial derivatives are
∂^2 f
∂x^2=12xy^2 ,∂^2 f
∂y^2=4x^3 +6y,∂^2 f
∂x∂y=12x^2 y,∂^2 f
∂y∂x=12x^2 y,the last two being equal, as expected.
5.2 The total differential and total derivativeHaving defined the (first) partial derivatives of a functionf(x, y), which give the
rate of change offalong the positivex-andy-axes, we consider next the rate of
change off(x, y) in an arbitrary direction. Suppose that we make simultaneous
small changes ∆xinxand ∆yinyand that, as a result,fchanges tof+∆f.
Then we must have
∆f=f(x+∆x, y+∆y)−f(x, y)=f(x+∆x, y+∆y)−f(x, y+∆y)+f(x, y+∆y)−f(x, y)=[
f(x+∆x, y+∆y)−f(x, y+∆y)
∆x]
∆x+[
f(x, y+∆y)−f(x, y)
∆y]
∆y.(5.3)In the last line we note that the quantities in brackets are very similar to those
involved in the definitions of partial derivatives (5.1), (5.2). For them to be strictly
equal to the partial derivatives, ∆xand ∆ywould need to be infinitesimally small.
But even for finite (but not too large) ∆xand ∆ythe approximate formula
∆f≈∂f(x, y)
∂x∆x+∂f(x, y)
∂y∆y, (5.4)