082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

830 Chapter 25 Functions of several variables
differentiating ∂z
∂y w.r.t.xproduces ∂ ( ) ∂z
∂x ∂y
differentiating ∂z
∂y w.r.t.yproduces ∂ ( ) ∂z
∂y ∂y
= ∂2 z
∂x∂y
= ∂2 z
∂y 2
For mostcommon functions, the mixed derivatives ∂2 z
∂y∂x and ∂2 z
∂x∂y areequal.
Example25.5 Given
z(x,y) = 3xy 3 −2xy +sinx
find allsecondpartialderivatives ofz.
Solution
∂z
∂x =3y3 −2y+cosx
∂ 2 z
∂x = ∂ ( ) ∂z
=−sinx
2 ∂x ∂x
∂ 2 z
∂x∂y = ∂ ( ) ∂z
=9y 2 −2
∂x ∂y
Note that
∂ 2 z
∂x∂y = ∂2 z
∂y∂x .
∂z
∂y = 9xy2 −2x
∂ 2 z
∂y = ∂ ( ) ∂z
= 18xy
2 ∂y ∂y
∂ 2 z
∂y∂x = ∂ ( ) ∂z
=9y 2 −2
∂y ∂x
Example25.6 Given
H(x,t) =3x 2 +t 2 +e xt
verify that ∂2 H
∂x∂t = ∂2 H
∂t∂x .
Solution
∂H
∂x =6x+text
∂ 2 H
∂t∂x = ∂ ∂t
∂ 2 H
∂x∂t = ∂ ∂x
∂H
∂t
=2t+xe xt
( ) ∂H
= ∂ ∂x ∂t (6x+text )= e xt +txe xt
( ) ∂H
= ∂ ∂t ∂x (2t+xext )=e xt +xte xt
Third,fourthandhigherderivativesarefoundinasimilarwaytofindingsecondderivatives.
The third derivatives are found by differentiating the second derivatives and
so on.