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z &ก x .
z &ก x .
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∂z
∂x=lim
h0
fxh , yf x , y
h
∂z
∂y=lim
h0
fx , yhf 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
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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
h0
f x1,, xih ,, xnfx1,, xi,, xn h
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f &ก x & ' &%. y % !" *3! ก &#
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(a, b)
∂ f
∂xa , b
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1.ก *+, f(x, y) = xy2exy fx(1, 1) 6 fy(0, 1)
2. f(x, y, z) = exyz(x – y)
3. " % &,$,&#-!*.35" x2 + 2y2 – z = 0 ก6
x = 1 * (1, -2, 9)
4. " % &,$,&#-!*.35" 2x2 + y2 + z2 = 4 ก 6 y = -1 * (1, -1, -1)
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+, 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*,"
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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
∂x⋅d x d t∂ f
∂y⋅d y d t
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+, 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$)%!"#6% กก" 4!"
∂z
∂s,∂z
∂t
∂z
∂s=∂ f
∂x⋅∂x
∂s∂ f
∂y⋅∂y
∂s ,
∂z
∂t=∂ f
∂x⋅∂x
∂t∂f
∂y⋅∂y
∂t
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f ก !"# & " &#-ก
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∂ f
∂x ,∂f
∂y
∂
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∂∂xf
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∂∂fy
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∂∂yf
,∂
∂x
∂∂fx
=∂∂2xf2, fx x, f11, D11f ∂∂y
∂∂xf
=∂∂y2∂fx, fx y, f1 2, D12f∂
∂x
∂∂fy
=∂∂x2∂fy, fy x, f21, D21f ∂∂y
∂∂fy
=∂∂2yf2, fy y, f2 2, D22f10
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ก *+, z = f(x, y) &#-ก !"# 2, x %"#.
∆x 6 y %"#. ∆y 61*," z # ก&*% &ก ∆z % & ก ∆z = f(x + ∆x, y + ∆y) - f(x, y)
2, z = f(x, y) %!&. & % & "% f *
(x, y) &*," df(x, y) ก *$*
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d fx , y=∂ f
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∂y d y
fxdx , ydy≈fx , ydf x , y
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