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

25.4 Higher order derivatives 831
Example25.7 Find all third derivatives of f (r,s) = sin(2r) −3r 4 s 2 .
Solution
∂f
∂r = 2cos(2r) −12r3 s 2
∂ 2 f
∂r 2 = −4sin(2r) −36r2 s 2
∂ 2 f
∂r∂s = −24r3 s
∂f
∂s = −6r4 s
∂ 2 f
∂s 2 = −6r4
The third derivatives are ∂3 f
∂r , ∂ 3 f
3 ∂r 2 ∂s , ∂ 3 f
∂r∂s and ∂3 f
, and these are found by differentiating
the second derivatives.
2 ∂s3 ∂ 3 f
∂r = ∂ ( ) ∂ 2 f
= −8cos(2r) −72rs 2
3 ∂r ∂r 2
∂ 3 f
∂r 2 ∂s = ∂ ( ∂ 2 )
f
= −72r 2 s
∂r ∂r∂s
∂ 3 f
∂r∂s = ∂ ( ∂ 2 )
f
= −24r 3
2 ∂r ∂s 2
∂ 3 f
∂s = ∂ ( ∂ 2 )
f
= 0
3 ∂s ∂s 2
Note that the mixed derivatives can be calculated inavariety ofways.
∂ 3 f
∂r 2 ∂s = ∂ ( ∂ 2 )
f
= ∂ ( ∂ 2 )
f
∂r ∂r∂s ∂s ∂r 2
∂ 3 f
∂r∂s = ∂ ( ) ∂ 2 f
= ∂ ( ) ∂ 2 f
2 ∂r ∂s 2 ∂s ∂r∂s
EXERCISES25.4
1 Calculateall second derivatives of v where
v(h,r) =r 2√ h
2 Find the second partialderivatives of f given
(a) f =x 2 y+y 3 (b) f =2x 4 y 3 −3x 3 y 5
(c) f=4 √ xy 2 (d) f = x2 +1
y
(e) f= 3x3 √ y
(f) f=4 √ xy
3 Find allsecond partialderivatives of
(a) z =xe 2y
(c) z =xcos(2x +3y)
(e) z = e x siny
(g) z = e xy
(b) z = 2sin(xy)
(d) z =ysin(4xy)
(f) z = e 3x−y
4 Find allsecond partialderivatives of
(a) z = (3x −2y) 20
(c) z = sin(x 2 +y 2 )
1
(e) z=
3x−2y
(b) z = √ 2x+5y
(d) z = ln(2x +5y)
5 Find allthirdpartialderivatives ofzwhere
z(x,y) =
x2
y +1
6 Evaluate all secondpartialderivatives of f atx = 2,
y=1.
(a) f =ye x (b) f =sin(2x −y)
( )
y
(c) f=ln
x