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อนุพันธยอย 1

2301116 ก II ก !"#

2

$

%

" %% & & '!

ก(ก$)

*

! %

& "%

3

%

+, z = f(x, y) &#-ก !"#

z &ก x .

z &ก x .

/ก0'.1*,ก

z

x=lim

h0

fxh , yf x , y

h

z

y=lim

h0

fx , yhf x , y

h

z

x, f

x , f

xx , y, fxx , y, Dxfx , y, D1f

z

y,f

y , f

yx , y, f yx , y, Dy fx , y, D2 f

4

n !"#

& % ก n !"#+ &*"ก

2, u = f(x1,x2,...,xn) & 1*," u &ก xi .

1. ก

f(x, y) = x/y

f(x, y) = 4y3ex + x ln y

u(x, y, z, w) = xy3z + zexw - w2zy

u

xi=lim

h0

f x1,, xih ,, xnfx1,, xi,, xn h

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5

" %% & & '!

f &ก x & ' &%. y % !" *3! ก &#

1# f 4&#&%.&,%5&,$,* (a, b) 6 y = b

(a, b)

f

xa , b

6

1.ก *+, f(x, y) = xy2exy fx(1, 1) 6 fy(0, 1)

2. f(x, y, z) = exyz(xy)

3. " % &,$,&#-!*.35" x2 + 2y2z = 0 ก6

x = 1 * (1, -2, 9)

4. " % &,$,&#-!*.35" 2x2 + y2 + z2 = 4 ก 6 y = -1 * (1, -1, -1)

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7

+, z = f(x, y) &#-ก !"#6 x = x(t) 6 y = y(t) &#- ก !"# t & 1*," z = f(x(t), y(t)) &#-ก !"# t &

ก & 1*,"

&ก ก(ก$)ก #6ก!"#

1. ก *+, z = xy – 2, x = et, y = 3t + 1 dz/dt

2. " %&:"+ก &.; ! % (x, y, z) $* z = xy &#- ก !"# x, y 6 x = t + 3, y = t2

ก(ก$)&%.%!"#6&4!"

d z d t= f

xd x d t f

yd y d t

8

+, z = f(x, y) &#-ก !"#6 x, y &#-ก !"

# s, t ก ". x = x(s, t) 6 y = y(s, t) &#-ก !"# s, t

& 1*," z = f(x(s, t), y(s, t)) &#-ก !"# s, t & ก &

' ก z &ก s 6 t1*,

3. ก *+, z = xy – 2, x = set, y = st + t + s

ก(ก$)%!"#6% กก" 4!"

z

s,z

t

z

s= f

xx

s f

yy

s ,

z

t= f

xx

tf

yy

t

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f ก !"# & " &#-ก

!"# *3& % 2 ' ก 31*, &ก"

* f )4%.

&

*

f

x ,f

y

x

fx

,y

xf

,x

fy

,y

yf

,

x

fx

=2xf2, fx x, f11, D11f

y

xf

=y2fx, fx y, f1 2, D12f

x

fy

=x2fy, fy x, f21, D21f

y

fy

=2yf2, fy y, f2 2, D22f

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&%." %%!"#%" %%ก&" t #/ &ก"ก5ก6

! ก &##!"#4%!! ก &##!"#.

&ก" #/ ! % ก ก,#/ ! % & + ,ก(ก$) "

1. ! ก &###% !ก6ก +'6=% " 4

3"6 10 3" 2, =%ก &%43*,"! 3 3"! 6" %

**,"! 2 3"!

2. !,1%,!,4&!$!+ก0'6&#-ก6ก 2, " %!,1%,&%

43+! 24 3"!#>6&,5 =ก &%43+! 3 3"!#>

" #% !!,1%,3&%43 *,"! & +* &%.!,1%,%" % 240 3"6&,5 =ก &#- 30 3"

! %

11

ก *+, z = f(x, y) &#-ก !"# 2, x %"&#1#.

x 6 y %"&#1#. y 61*," z &#1# ก&*% &ก z % & ก z = f(x + x, y + y) - f(x, y)

2, z = f(x, y) %!&. & % & "% f *

(x, y) &*," df(x, y) ก *$*

& % 2#6% ' " % *&.1*,$*

& "%

d fx , y= f

x d xf

y d y

fxdx , ydy≈fx , ydf x , y

12

1. & "%ก !1#3

2. ก"ก%+4&## ก=% 3 3"6 4 3"1#&#- =% 3.03

3"6 3.9 3" #6% '"&###% !ก"

+3 $*+ , & "%

!" ก & "%

f x , y=lnx2y2 f x , y=x2yxy2 f x , y , z=

x2y2z2

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