𝑧(𝑥, 𝑦) =
𝑍(𝑥, 1) =
𝑥𝑠𝑖𝑛(𝑥 − 1)
2 + 𝑥2
=≫ 𝐹 ′ 𝑥 (1,1,0) =
𝑍(1, 𝑦) =
𝑥𝑦𝑠𝑖𝑛(𝑥 − 𝑦)
, (1,1,0)
1 + 𝑥 2 +𝑦 2
𝜕𝑧 ((sin⁡(𝑥 − 1) + 𝑥𝑐𝑜𝑠(𝑥 − 1)(2 + 𝑥 2 ) − 2𝑥 2 (sin⁡(𝑥 − 1) 1
=
=
(2 + 𝑥 2 )2
𝜕𝑥
3
𝑦𝑠𝑖𝑛(1 − 𝑦)
2 + 𝑦2
=≫ 𝐹 ′ 𝑦 (1,1,0) =
𝐹 ′ 𝑧 (1,1,0) =
𝜕𝑧 ((sin⁡(1 − 𝑦) − 𝑦𝑐𝑜𝑠(1 − 𝑦)(2 + 𝑦 2 ) − 2𝑦 2 (sin⁡(1 − 𝑦)
1
=
=−
2
2
(2 + 𝑦 )
𝜕𝑦
3
𝜕𝑧
=1
𝜕𝑥
We have the tangent plane:
𝐹 ′ 𝑧 (𝑥, 𝑦, 𝑧)(𝑧 − 𝑧0 ) = 𝐹 ′ 𝑥 (𝑥, 𝑦, 𝑧)(𝑥 − 𝑥0 ) + 𝐹 ′ 𝑦 (𝑥, 𝑦, 𝑧)(𝑦 − 𝑦0 )
=≫ 1(𝑧 − 0) =
1
1
(𝑥 − 1) − (𝑦 − 𝑦0 )
3
3
1
1
=≫ 𝑧 = 𝑥 − 𝑦
3
3