Understanding Engineering Mathematics

16.4 Higher order derivatives

Ifz=f(x,y)is a function ofxandy,then

∂z ∂x

is also a function ofxandyand so

may itself be partially differentiated with respect tox,

∂ ∂x

( ∂z ∂x

) or with respect toy, ∂ ∂y

( ∂z ∂x

)

. We write suchsecond order derivatesas:

∂^2 z ∂x^2

,

∂^2 z ∂x∂y

,

∂^2 z ∂y∂x

,

∂^2 z ∂y^2

which we will sometimes write aszxx,zxy,zyx,zyy, respectively.
For all functions we are interested in we may assume that

∂^2 z ∂x∂y

=

∂^2 z ∂y∂x

or zxy=zyx

Similarly, higher order derivates may be defined, such as

∂^3 z ∂x^2 ∂y

=zxxy,etc.

Problem 16.2 Evaluate all second order derivatives of the function

f.x,y/=x^3 Yy^3 Y 2 xyYx^2 y

and verify that

@^2 f @x@y

=

@^2 f @y@x

The differentiation off(x,y)=x^3 +y^3 + 2 xy+x^2 yis routine and we obtain

fx= 3 x^2 + 2 y+ 2 xy fxx= 6 x+ 2 y fy= 3 y^2 + 2 x+x^2 fyy= 6 y

fxy=(fx)y=( 3 x^2 + 2 y+ 2 xy)y= 2 + 2 x

fyx=(fy)x=( 3 y^2 + 2 x+x^2 )x= 2 + 2 x =fxy

Exercises on 16.4

Find all first and second order partial derivatives of the following functions,f(x,y),
checking the equality of the mixed derivatives

(i) f(x,y)=x^3 y^2 + 4 xy^4 (ii) f(x,y)=exycos(x+y)