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

25.4 Higher order derivatives 829
∂z
2 (a)
∂x = 1 y , ∂z
∂y = − x y 2
y
(b)
2 √ x , √ x
(c) − 3y2
x 4 , 2y
x 3
(d) − 1
x 2 y ,− 1
xy 2
(e)
(f)
√
6x y
y + x 2 , −3x2 y 2 − 1
2x √ y
1
2√ y
x −3, 1 2
√ x
y −3
∂z ∂z
3 (a) =2cosx,
∂x ∂y =−3siny
(b) siny,xcosy
(c) tany −y 2 cosx,xsec 2 y −2ysinx
(d) ysiny,xsiny+xycosy
(e) cos(x +y),cos(x +y)
(f) −16sin(4x −6y),24sin(4x −6y)
(g) − siny
x 2 , cosy
x
∂y
4 (a)
∂x =tex , ∂y
∂t = ex
(b) 2xe −t ,−x 2 e −t
(c) 1,e t (1+t)
(d) 6xe 2t −t 3 e −x ,6x 2 e 2t +3t 2 e −x
(e) te xt ,xe xt
(f) 2e 2x+3t ,3e 2x+3t
5 (a)
(b)
(c)
∂z
∂x =
x ∂z
√
x 2 +y2, ∂y =
3y
(2x +3y) 2 , − 3x
(2x +3y) 2
( ( )
1
y cos x
), − x y y 2 cos x
y
(d) 3ye 3xy ,3xe 3xy
(e)
(f)
2
2x−3y ,− 3
2x−3y
1
x , 1 y
(g) 1 +ln(xy), x y
( )
y
(h) ln −1, x x y
y
√
x 2 +y 2
∂f ∂f
6 (a) (1,2) =8,
∂x ∂y (1,2)=1
(b) 0.5, −0.25
7
(c) 4.5234, 3.0156
(d) 29.5562, 14.7781
(e) 3,0.5
(f) 0.6708, 0.2236
∂f
∂f
(2,1) = 7.6464, (2,1) = 6.5858
∂r ∂h
25.4 HIGHERORDERDERIVATIVES
Just as functions of one variable have second and higher derivatives, so do functions of
several variables. Consider
z =z(x,y)
Thefirstpartialderivativesofzare ∂z ∂z
and
∂x ∂y .Thesecondpartialderivativesarefound
bydifferentiatingthefirstderivatives.Wecandifferentiatefirstpartialderivativeseither
w.r.t.xor w.r.t.ytoobtain various second partialderivatives:
differentiating ∂z
∂x w.r.t.xproduces ∂ ( ) ∂z
∂x ∂x
differentiating ∂z
∂x w.r.t.yproduces ∂ ( ) ∂z
∂y ∂x
= ∂2 z
∂x 2
= ∂2 z
∂y∂x