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|f(ρ cos θ, ρ sin θ)| =
arctan ρ
2
ρ2
ρ → 0,
f(x, y) = f(y, x)
lim
t→0 +
f(t, 0) − f(0, 0)
=
t
arctan t2
t2 ∂xf(0, 0) = ∂yf(0, 0) = 1 L L(x, y) = x+y
f(x,
y) − f(0, 0) − L(x, y)
x2 + y2 =
arctan(ρ
2
)
ρ − ρ(cos θ + sin θ)
ρ
=
− (cos θ + sin θ)
θ = π/4
lim
f(x,
y) − f(0, 0) − L(x, y)
= lim
x2 + y2
(x,y)→(0,0)
y=x
(0, 0)
arctan(ρ 2 )
ρ 2
ρ→0 +
= 1,
arctan(ρ 2 )
ρ 2
− √
2
→ √ 2 − 1 = 0.
f(x, y) = y 2/3 (y + x 2 − 1) ∂yf
f(x, y) = 3 x 2 (y − 1)+1 (0, 1)
Dvf(0, 1) v
R 2
f(x, y) =
x 2 y
0
arctan t
t
y = 0 ∂yf(x, y) = 2
3 3√ y (y + x2 − 1) + y 2/3 y = 0
dt.
f(x, y) − f(x, 0)
∂yf(x, 0) = lim
= lim
y→0 y
y→0
y + x2 − 1
√ .
3 y
x 2 − 1 = 0 x = ±1
∂yf(±1, 0) = 0
(0, 1) x = ρ cos θ y = ρ sin θ + 1
f(ρ cos θ, ρ sin θ + 1) = 3 ρ 3 cos θ sin θ = ρ 3√ cos θ sin θ.
(0, 1) ρ = 0 f(0, 1) = 1 v v = (cos θ, sin θ)
∂vf(0, 1) = lim
t→0 +
f((0, 1) + tv) − f(0, 1)
=
t
3√ cos θ sin θ.
v ↦→ ∂vf(0, 1)