• Author / Uploaded
• Omar McBell

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1. a) 𝑓(𝑥, 𝑦) = 𝑥 2 + 5𝑦 + 1 𝑓(𝑥) = 2𝑥 𝑓(𝑦) = 5 b) 𝑓 = 𝑥 3 + 3√𝑦 𝑓(𝑦) = 3𝑥 2 1 1 3 2 1 − 3 . 𝑦 3 1

𝑓(𝑦) = . 𝑦 3−1 𝑓(𝑦) = 𝑓(𝑦) =

3

3 √𝑦 2

c) 𝑓 = 𝐿𝑛𝑥 + +𝑒 𝑦 1

𝑓(𝑥) = 𝑥

𝑓(𝑥) = 𝑒 𝑦 . 1

2. a) 𝑓(𝑥, 𝑦) = 𝑥 4 + 𝑦 9 𝑓(𝑥) = 4𝑥 3 𝑓(𝑦) = 9𝑦 8 b)𝑓(𝑥, 𝑦) = 2𝑥 7 + 3𝑦 5 − 5𝑥 𝑓(𝑥) = 14𝑥 6 − 5 𝑓(𝑦) = 15𝑦 4 c) 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑥𝑦 + 𝑦 6 𝑓(𝑥) = 3𝑥 2 + 𝑦 𝑓(𝑦) = 𝑥 + 6𝑦 5 d)𝑓(𝑥, 𝑦) = 2𝑥 + 3𝑦 + 1 𝑓(𝑥) = 2𝑥 . 1. 𝑙𝑛2 𝑓(𝑦) = 3𝑦 . 1. 𝑙𝑛3 e) 𝑓(𝑥, 𝑦) = 𝑥𝑒 𝑦 + 𝑦𝑒 𝑥 + 𝑒 𝑓(𝑥) = 𝑒 𝑦 + 𝑒 𝑥 . 𝑦. 1 𝑓(𝑦) = 𝑥. 𝑒 𝑦 . 1 + 𝑒 𝑥 f) 𝑓(𝑥, 𝑦) = 𝑆𝑒𝑛𝑥. 𝐶𝑜𝑠𝑦 𝑓(𝑥) = 1. 𝐶𝑜𝑠𝑥 𝑓(𝑦) = −1. 𝑆𝑒𝑛𝑦

g) 𝑓(𝑥, 𝑦) = 𝑆𝑒𝑛(𝑥 8 − 𝑦 5 ) 𝑓(𝑥) = (𝑥 8 − 𝑦 5 )´ 𝐶𝑜𝑠(𝑥 8 − 𝑦 5 ) 𝑓(𝑥) = 8𝑥 7 𝐶𝑜𝑠(𝑥 8 − 𝑦 5 ) 𝑓(𝑦) = (𝑥 8 − 𝑦 5 )´ 𝐶𝑜𝑠(𝑥 8 − 𝑦 5 ) 𝑓(𝑦) = −5𝑦 4 𝐶𝑜𝑠(𝑥 8 − 𝑦 5 ) 3

h) 𝑓(𝑥, 𝑦) = 𝑒 2𝑥+𝑦 + 𝑒 𝑥𝑦 + 𝑒 2 3 𝑓(𝑥) = 𝑒 2𝑥+𝑦 . (2𝑥 + 𝑦 3 )´ + 𝑒 𝑥𝑦 . (𝑥𝑦)´ 3 𝑓(𝑥) = 𝑒 2𝑥+𝑦 (2) + 𝑒 𝑥𝑦 (𝑥´. 𝑦 + 𝑦´. 𝑥) 3 𝑓(𝑥) = 2𝑒 2𝑥+𝑦 + 𝑦𝑒 𝑥𝑦 3 𝑓(𝑦) = 𝑒 2𝑥+𝑦 (2𝑥 + 𝑦3)´ + 𝑒 𝑥𝑦 (𝑥𝑦)´ 3 𝑓(𝑦) = 𝑒 2𝑥+𝑦 (2𝑥 + 𝑦 3 )´ + 𝑒 𝑥𝑦 (𝑥𝑦)´ 3 𝑓(𝑦) = 𝑒 2𝑥+𝑦 (3𝑦 2 ) + 𝑒 𝑥𝑦 (𝑥. ý + 𝑦´. 𝑥) 3 𝑓(𝑦) = 3𝑦 2 . 𝑒 2𝑥+𝑦 + 𝑒 𝑥𝑦 . 𝑥 i) 𝑓(𝑥, 𝑦, 𝑧) = 𝑥 5 + 5𝑦 + 𝑆𝑒𝑛𝑧 − 𝑥𝑦𝑧 𝑓(𝑥) = 5𝑥 4 − 𝑦𝑧 𝑓(𝑧) = 5𝑦 . 1. 𝑙𝑛5 − 𝑥𝑧 𝑓(𝑧) = 1. 𝐶𝑜𝑠𝑧 − 𝑥𝑦

3. a) 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑦 5 + 𝑥 7 − 𝑦 2 ; (2,3) 𝑓(𝑥) = 3𝑥 2 + 7𝑥 6 → 𝑥 = 2 → 460 𝑓(𝑦) = 5𝑦 4 − 2𝑦 → 𝑦 = 3 → 399 b)𝑓(𝑥, 𝑦) = 𝑥 2 + 𝑥𝑦 3 − 3𝑥; (2,1) 𝑓(𝑥) = 2𝑥𝑦 + 𝑦 3 − 3 → 1 𝑓(𝑦) = 𝑥 2 + 3𝑥𝑦 2 → 10 c) 𝑓(𝑥, 𝑦) = (𝑥 2 − 5)5 𝑦 5 + 𝑥 2 (𝑦 5 + 1)2 ; (1,1) 𝑓(𝑥) = 𝑦 5 [5(𝑥 2 − 1)4 ] + (𝑦 5 + 1)2 (2𝑥) 𝑓(𝑦) = (𝑥 2 − 1)5 𝑦 5 +𝑥 2 (𝑦 5 + 1)2 ; (1,1) 𝑓(𝑦) = 5𝑦 4 (𝑥 2 − 1)5 + 𝑥 2 [2(𝑦 5 + 1)1 (5𝑦 4 )]; (1,1) d) 𝑓(𝑥, 𝑦) = 3𝑥 + 𝑒 𝑦 − 𝑥 2 𝑦 3 + 2𝑥𝑦 ; (0,0) 𝑓(𝑥) = 3𝑥 𝑙𝑛3 − 2𝑥𝑦 3 + 2𝑥𝑦 𝑦𝑙𝑛2(0,0)

4. a) 𝑢 = 𝑥 3 + 𝑦 3 → 𝑥𝑢𝑥 + 𝑦𝑢𝑦 = 3𝑢 𝑥(3𝑥 2 ) + 𝑦(3𝑦 2 ) = 3(𝑥 3 + 𝑦 3 ) 3𝑥 3 + 3𝑦 3 = 3(𝑥 3 + 𝑦 3 ) 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑦 3 𝑓(𝑥) = 3𝑥 2 𝑓(𝑦) = 3𝑦 2 b) 𝑢 = √𝑥 2 + 𝑦 2 → 𝑥𝑢𝑥 + 𝑦𝑢𝑦 = 3𝑢 𝑥(𝑥) √𝑥 2 +𝑦 2

+

𝑦(𝑦) √𝑥 2 +𝑦 2 𝑥 2 +𝑦 2

= 3√𝑥 2 + 𝑦 2

√𝑥 2 +𝑦 2

= 3√𝑥 2 + 𝑦 2 1

(𝑥 2 + 𝑦 2 )1−2 1

(𝑥 2 + 𝑦 2 )2 ≠ 3√𝑥 2 + 𝑦 2 𝑓(𝑥, 𝑦) = √𝑥 2 + 𝑦 2 1

𝑓(𝑥) = (𝑥 2 + 𝑦 2 )2 1

1

𝑓(𝑥) = 2 (𝑥 2 + 𝑦 2 )−2 . 2𝑥 𝑓(𝑥) =

𝑥

√𝑥 2 +𝑦2 1

𝑓(𝑦) = (𝑥 2 + 𝑦 2 )2 1 2

1

𝑓(𝑦) = (𝑥 2 + 𝑦 2 )−2 . 2𝑦 𝑓(𝑦) =

𝑦

√𝑥 2 +𝑦2

5. a) 𝑓 = 𝑥 5 + 𝑦 3 − 𝑥 2 𝑦 4 ; 𝑓𝑥𝑥 ; 𝑓𝑦𝑦𝑦 𝑓𝑥 = 5𝑥 4 − 2𝑥𝑦 4 𝑓𝑥𝑥 = 20𝑥 3 − 2𝑦 4 𝑓𝑦 = 3𝑦 2 − 4𝑥 2 . 𝑦 3 𝑓𝑦𝑦 = 6𝑦 − 12𝑥 2 𝑦 2 𝑓𝑦𝑦𝑦 = 6 − 24𝑥 2 b) 𝑓 = 𝑥 2 𝑦 − 𝑥𝑦 3 − 𝑥𝑦; 𝑓𝑥𝑦 ; 𝑓𝑦𝑥 𝑓𝑥 = 2𝑥𝑦 − 𝑦 3 − 𝑦 𝑓𝑥𝑦 = 2𝑥 − 3𝑦 2 − 1 𝑓𝑦 = 𝑥 2 − 3𝑥𝑦 2 − 𝑥 𝑓𝑦𝑥 = 2𝑥 − 3𝑦 2 − 1

c) f = 𝑥 2 𝑦 3 𝑧 4 ; 𝑓𝑥𝑦𝑧 ; 𝑓𝑥𝑥𝑦𝑧 𝑓𝑥 = 2x𝑦 3 𝑧 4 𝑓𝑥𝑦 = 6x𝑦 2 𝑧 4 𝑓𝑥𝑦𝑧 = 24x𝑦 2 𝑧 3 𝑓𝑥 = 2x𝑦 3 𝑧 4 𝑓𝑥𝑥 = 2𝑦 3 𝑧 4 𝑓𝑥𝑥𝑦 = 6𝑦 2 𝑧 4 𝑓𝑥𝑥𝑦𝑧 = 24𝑦 2 𝑧 3 d) 𝑓 = 𝑒 𝑥𝑦 -1 ; 𝑓𝑥𝑦 ; 𝑓𝑥𝑥𝑦 𝑓𝑥 = 𝑒 𝑥𝑦 . y 𝑓𝑥𝑦 = 𝑒 𝑥𝑦 (y)´ + y(𝑒 𝑥𝑦 )´ 𝑓𝑥𝑦 = 𝑒 𝑥𝑦 + y. 𝑒 𝑥𝑦 . 𝑥 𝑓𝑥𝑥 = y. 𝑒 𝑥𝑦 y 𝑓𝑥𝑥𝑦 = 𝑦 2 . 𝑒 𝑥𝑦 𝑓𝑥𝑥𝑦 = 𝑦 2 (𝑒 𝑥𝑦 )´ + 𝑒 𝑥𝑦 (𝑦 2 )´ 𝑓𝑥𝑥𝑦 = 𝑦 2 𝑒 𝑥𝑦 𝑥 + 𝑒 𝑥𝑦 . 2y e) 𝑓 = 𝑥 𝑦𝑧 + 1; 𝑓𝑦𝑧 ; 𝑓𝑥𝑥 𝑓𝑦 = 𝑥 𝑦𝑧 . 𝑧𝑙𝑛𝑥 𝑓𝑦𝑧 = 𝑙𝑛𝑥[𝑥 𝑦𝑧 (𝑧)´ + 𝑧(𝑥 𝑦𝑧 )´] 𝑓𝑦𝑧 = 𝑙𝑛𝑥[𝑥 𝑦𝑧 + 𝑧. 𝑥 𝑦𝑧 . 𝑦. 𝑙𝑛𝑥] 𝑓𝑥 = 𝑦𝑧𝑥 𝑦𝑧−1 𝑓𝑥𝑥 = (𝑦𝑧 − 1)𝑦𝑧. 𝑥 𝑦𝑧−2 f) 𝑓 = 𝑒 3𝑥 + 𝑆𝑒𝑛(2𝑦) + 𝑙𝑛(7𝑧); 𝑓𝑥𝑥𝑥𝑥 ; 𝑓𝑦𝑦𝑦𝑦 ; 𝑓𝑧𝑧𝑧𝑧 𝑓𝑥 = 𝑒 3𝑥 . 3 𝑓𝑥𝑥 = 3. 𝑒 3𝑥 . 3 𝑓𝑥𝑥𝑥 = 9. 𝑒 3𝑥 . 3 𝑓𝑥𝑥𝑥𝑥 = 27. 𝑒 3𝑥 . 3 𝑓𝑦 = 2𝑐𝑜𝑠2 𝑓𝑦𝑦 = 2(−2sen. 2y) 𝑓𝑦𝑦 = −4Sen(2y) 𝑓𝑦𝑦𝑦 = −4[2𝐶𝑜𝑠2𝑦] 𝑓𝑦𝑦𝑦 = −8 cos(2y) 𝑓𝑦𝑦𝑦𝑦 = −8[−2𝑆𝑒𝑛2𝑦] 𝑓𝑦𝑦𝑦𝑦 = 16Sen2y

7

1

𝑓𝑧 = 7𝑧 = 𝑧 𝑓𝑧𝑧 = 𝑓𝑧𝑧 = 𝑓𝑧𝑧 =

𝑧(1)´−1(𝑧)´ 𝑧2 0−1 𝑧2 −1 𝑧2 𝑧(−1)´−(−1)(𝑧 2 )´

𝑓𝑧𝑧𝑧 =

𝑓𝑧𝑧𝑧𝑧 = 𝑓𝑧𝑧𝑧𝑧 =

6.

(𝑧 2 )2 𝑧 3 (2)´−2(𝑧 3 )´

=

2 𝑧3

(𝑧 5 )2 6 − 𝑧4

a) 𝑢(𝑥,𝑦) = 𝑥 2 + 𝑦 3 ; 𝑥(𝑡) = 3t + 1 ; 𝑦(𝑡) = 𝑡 5 ; 𝑢𝑡 𝑑𝑥 = 3t + 𝑑𝑡 𝑑𝑦 = 𝑡5 𝑑𝑡

1

𝑢(𝑡) = (3𝑡 + 1)2 + (𝑡 5 )3 𝑢(𝑡) = 2(3t + 1)(3) + 3(𝑡 5 )2 (5𝑡 4 ) 𝑢(𝑡) = 6(3t + 1) + 15𝑡14

b) 𝑢(𝑥,𝑦) = 3𝑥 5 + 2𝑦 3 ; 𝑥(𝑡,𝑠) = 𝑡 2 + 𝑠 4 ; 𝑦(𝑡,𝑠) = 3ts ; 𝑢𝑡 , 𝑢𝑠 𝑢(𝑡,𝑠) = 3(𝑡 2 + 𝑠 4 )5 + 2(3𝑡𝑠)3 𝑢(𝑡) = 3.5. (𝑡 2 𝑠 4 )5 (2t) + 2.3(3𝑡𝑠)3 (35) 𝑢(𝑡) = 30t. (𝑡 2 𝑠 4 )4 + 18s(3𝑡𝑠)2 𝑢(𝑠) = 3(𝑡 2 + 𝑠 4 )5 + 2(3𝑡𝑠)3 𝑢(𝑠) = 15(𝑡 2 + 𝑠 4 )5 (4𝑠 3 ) + 6(3𝑡𝑠)2 (3t) 𝑢𝑠 = 60𝑠 3 (𝑡 2 + 𝑠 4 )4 + 18t(3𝑡𝑠)2

7. a) 𝑧 4 − 𝑥 5 𝑦 7 − 1 = x + y 𝑧4 = x + y + 1 + 𝑥5𝑦7 4 z = √𝑥 + 𝑦 + 1 + 𝑥 5 𝑦 7 1

z = (𝑥 + 𝑦 + 1 + 𝑥 5 𝑦 7 )4 3

1

𝑧𝑥 = 4 (𝑥 + 𝑦 + 1 + 𝑥 5 𝑦 7 )−4 (1 + 5𝑥 4 𝑦 7 ) 𝑧𝑥 = 𝑧𝑦 = 𝑧𝑦 =

1+5𝑥 4 𝑦 7

4

4 √(𝑥+𝑦+1+𝑥 5 𝑦 7 )3 3 1 5 7 −4 (𝑥 + 𝑦 + 1 + 𝑥 𝑦 ) 4 1+7𝑥 5 𝑦 6

(1 + 7𝑦 6 𝑥 5 )

4

4 √(𝑥+𝑦+1+𝑥 5 𝑦 7 )3

b) 𝑧 5 𝑥 3 + 𝑧 2 𝑦 4 + 𝑦 3 + 𝑥 4 = 𝑥𝑦 𝑧 2 (𝑧 3 𝑥 3 +𝑦 4 ) + 𝑦 3 + 𝑥 4 = 𝑥𝑦 𝑧2 =

𝑥𝑦−𝑦 3 −𝑥 4 𝑧 3 𝑥 3 +𝑦 4 1

𝑧=

𝑥𝑦−𝑦 3 −𝑥 4 2 ( 𝑧3 𝑥 3 +𝑦4 )

1 𝑥𝑦 − 𝑦 3 − 𝑥 4 𝑧𝑥 = ( 3 3 ) 2 𝑧 𝑥 + 𝑦4 1 𝑥𝑦 − 𝑦 3 − 𝑥 4 𝑧𝑥 = ( 3 3 ) 2 𝑧 𝑥 + 𝑦4 1 𝑥𝑦 − 𝑦 3 − 𝑥 4 𝑧𝑥 = ( 3 3 ) 2 𝑧 𝑥 + 𝑦4

1 − 2

´

.( 1 − 2

. 1 − 2

𝑥𝑦 − 𝑦 3 − 𝑥 4 ) 𝑧3𝑥3 + 𝑦4

(𝑧 3 𝑥 3 + 𝑦 4 )(𝑥𝑦 − 𝑦 3 − 𝑥 4 ) − (𝑥𝑦 − 𝑦 3 − 𝑥 4 )(𝑧 3 𝑥 3 + 𝑦 4 ) (𝑧 3 𝑥 3 + 𝑦 4 )2

(𝑧 3 𝑥 3 + 𝑦 4 )(𝑦 − 4𝑥 3 ) − (𝑥𝑦 − 𝑦 3 − 𝑥 4 )(3𝑥 2 𝑧 3 ) . (𝑧 3 𝑥 3 + 𝑦 4 )2

1

1 𝑥𝑦 − 𝑦 3 − 𝑥 4 2 𝑧 3 𝑥 3 − 4𝑥 6 𝑧 3 + 𝑦 5 − 4𝑥 3 𝑦 4 − 3𝑥 3 𝑦𝑧 3 + 3𝑥 2 𝑦 3 𝑧 3 + 3𝑥 6 𝑧 3 𝑧𝑥 = ( 3 3 ) ( ) (𝑧 3 𝑥 3 + 𝑦 4 )2 2 𝑧 𝑥 + 𝑦4 1

1 𝑥𝑦 − 𝑦 3 − 𝑥 4 2 −2𝑥 3 𝑦𝑧 3 − 𝑥 6 𝑧 3 + 𝑦 5 − 4𝑥 3 𝑦 4 − 3𝑥 3 𝑦 3 𝑧 3 𝑧𝑥 = ( 3 3 ) ( ) (𝑧 3 𝑥 3 + 𝑦 4 )2 2 𝑧 𝑥 + 𝑦4

c) 𝑧 3 − 𝑆𝑒𝑛(𝑥 2 + 𝑦𝑧) = 1 𝑧 3 = 1 + 𝑆𝑒𝑛(𝑥 2 + 𝑦𝑧) −

𝑧 = (1 + 𝑠𝑒𝑛(𝑥 2 + 𝑦𝑧))

1 3

−

1

2

𝑧𝑥 = 3 (1 + 𝑆𝑒𝑛(𝑥 2 + 𝑦𝑧)) 3 (𝐶𝑜𝑠(𝑥 2 + 𝑦𝑧)2𝑥) 𝑧𝑥 =

2𝐶𝑜𝑠(𝑥 2 +𝑦𝑧) 3

2

3 √(1+𝑆𝑒𝑛(𝑥 2 +𝑦 2 ))

−

1

2

𝑧𝑦 = 3 (1 + 𝑆𝑒𝑛(𝑥 2 + 𝑦𝑧)) 3 (𝐶𝑜𝑠(𝑥 2 + 𝑦𝑧)𝑧) 𝑧𝑦 =

𝑧𝑐𝑜𝑠(𝑥 2 +𝑦𝑧) 3

3 √1+𝑆𝑒𝑛(𝑥 2 +𝑦𝑧)2

8. a) 𝑧 6 − 𝑥 5 − 𝑦 4 = 1; 𝑍𝑥𝑥 ; 𝑍𝑦𝑦 𝑧6 = 1 + 𝑥5 + 𝑦4 1

𝑧 = (1 + 𝑥 5 + 𝑦 4 )6 5

1

𝑧𝑥 = 6 (1 + 𝑥 5 + 𝑦 4 )−6 (5𝑥 4 ) →

5𝑥 4 (1 + 6

5

𝑥 5 + 𝑦 4 )−6

´ 5 5 ´ 5𝑥 4 5𝑥 4 (1 + 𝑥 5 𝑦 4 )−6 + ( ) ((1 + 𝑥 5 𝑦 4 )−6 ) ) 6 6 5 11 4 20𝑥 3 5𝑥 5 − (1 + 𝑥 5 𝑦 4 ) 6 + ( ) (− (1 + 𝑥 5 𝑦 4 )− 6 . 5𝑥 4 ) 6 6 6 5 11 20 3 125 8 − 5 4 𝑥 (1 + 𝑥 𝑦 ) 6 − 𝑥 (1 + 𝑥 5 𝑦 4 )− 6 6 36 5 5 3 25 − 5 4 6 𝑥 (1 + 𝑥 𝑦 ) [4 − 6 𝑥 5 (1 + 𝑥 5 𝑦 4 )−1 ] 6

𝑧𝑥𝑥 = ( 𝑧𝑥𝑥 = 𝑧𝑥𝑥 = 𝑧𝑥𝑥 =

1

𝑧 = (1 + 𝑥 5 𝑦 4 )6 5 1 6 5 4 𝑧𝑦 = 6 𝑦 3 (1 + 𝑥 5 + 𝑦 4 )−6 5 11 12 4 5 𝑧𝑥𝑥 = 6 𝑦 2 (1 + 𝑥 5 + 𝑦 4 )−6 + 6 𝑦 3 [− 6 (1 + 𝑥 5 + 𝑦 4 )− 4 4𝑦 3 ] 5 11 12 80 𝑧𝑦𝑦 = 6 𝑦 2 (1 + 𝑥 5 + 𝑦 4 )−6 + 36 𝑦 6 (1 + 𝑥 5 + 𝑦 4 )− 6

𝑧𝑦 = (1 + 𝑥 5 + 𝑦 4 )−6 (4𝑦 3 )

b) 𝑧 3 − 𝑥 2 𝑦 4 = 1 ; 𝑧𝑥𝑦 𝑧3 = 1 + 𝑥2𝑦4 1

𝑧 = (1 + 𝑥 2 𝑦 4 )3 2

1

𝑧𝑥 = 3 (1 + 𝑥 2 𝑦 4 )−3 (2𝑥𝑦 4 ) 2

2

𝑧𝑥 = 3 𝑥𝑦 4 (1 + 𝑥 2 𝑦 4 )−3 8

2

2

8

2

16 3 7 𝑥 𝑦 (1 + 9

5

2

𝑧𝑥𝑦 = 3 𝑥𝑦 3 (1 + 𝑥 2 𝑦 4 )−3 + 3 𝑥𝑦 4 [− 3 (1 + 𝑥 2 𝑦 4 )−3 4𝑥 2 𝑦 3 ] 𝑧𝑥𝑦 = 3 𝑥𝑦 3 (1 + 𝑥 2 𝑦 4 )−3 +

9. 5.1) 𝑧 = 𝑥 2 𝑦 3 + 2𝑥𝑦 𝑧𝑥 = 2𝑥𝑦 3 + 2𝑦 𝑧𝑦 = 3𝑥 2 𝑦 2 + 2𝑥

𝑥 𝑦2 𝑥 ´

5.2) 𝑧 =

−

𝑦 𝑥2 𝑦 ´

𝑧𝑥 = (𝑦2 ) − (𝑥 2 ) ´

𝑧𝑥 =

𝑦 2 (𝑥)´−𝑥(𝑦 2 ) (𝑦 2 )2 𝑦2

𝑧𝑥 = 𝑦4 + 1

𝑧𝑥 = 𝑦2 +

´

−(

𝑧𝑦 = 𝑧𝑦 =

𝑥 2 (𝑥)´ −𝑦(𝑥 2 ) ) (𝑥 2 )2

2𝑥𝑦 𝑥4 2𝑦 𝑥3 ´

𝑧𝑦 =

𝑦 2 (𝑥)´−𝑥(𝑦 2 ) (𝑦 2 )2 −2𝑥𝑦 𝑥2 − 𝑦4 𝑥4 2𝑥 1 − 𝑦3 − 𝑥 2

−

5

𝑥 2 𝑦 4 )−3

𝑥 2 (𝑦)´−𝑦(𝑥 2 ) (𝑥 2 )2

5.3) 𝑧 = 𝑆𝑒𝑛2𝑥𝐶𝑜𝑠4𝑥𝑦 𝑧𝑥 = (𝑆𝑒𝑛2𝑥)´(𝐶𝑜𝑠4𝑥𝑦) + (𝑆𝑒𝑛2𝑥)(𝐶𝑜𝑠4𝑥𝑦)´ 𝑧𝑥 = 2𝐶𝑜𝑠𝑥𝐶𝑜𝑠4𝑥𝑦 − 4𝑦𝑆𝑒𝑛2𝑥𝑆𝑒𝑛4𝑥𝑦 𝑧𝑦 = (𝑆𝑒𝑛2𝑥)´(𝐶𝑜𝑠4𝑥𝑦) + (𝑆𝑒𝑛2𝑥)(𝐶𝑜𝑠4𝑥𝑦)´ 𝑧𝑦 = 𝑆𝑒𝑛2𝑥(−4𝑥𝑆𝑒𝑛4𝑥𝑦) 𝑧𝑦 = −4𝑥𝑆𝑒𝑛2𝑥𝑆𝑒𝑛4𝑥𝑦

𝑥

5.4) 𝑧 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑦) 𝑧𝑥 =

1 𝑦

( ) 𝑥 2 𝑦

1+( )

𝑧𝑥 =

1 𝑦 𝑥2 1+ 2 𝑦

𝑧𝑥 =

𝑦 𝑦 2 +𝑥 2

𝑧𝑦 = 𝑧𝑦 = 𝑧𝑦 =

1 𝑦 𝑦2 +𝑥2 𝑦2

=

𝑥 𝑦

( ) 𝑥 2 𝑦 𝑦(𝑥)´−𝑥(𝑦)´ 𝑦2 𝑥2 1+ 2 𝑦 −𝑥 𝑦2 𝑦2 +𝑥2 𝑦2

1+( )

−𝑥

𝑧𝑦 = 𝑦2 +𝑥 2

𝑥

𝑦

5.5) 𝑧 = 𝑒 𝑦 + 𝑒 𝑥 𝑥

𝑧𝑥 =

𝑥 𝑦 𝑥 1 𝑒 ⁄𝑦 𝑦

𝑧𝑥 =

𝑒𝑦 𝑦

𝑦

𝑦 𝑥

𝑧𝑥 = 𝑒 𝑦 ( ) + 𝑒 𝑥 ( ) +𝑒

𝑥

𝑥 𝑦

𝑦⁄ −𝑦 𝑥( ) 𝑥2

𝑦

−

𝑦𝑒 𝑥 𝑥2

𝑥

𝑦

−𝑥 𝑦2

𝑒𝑥1 𝑥

𝑦

𝑧𝑦 = 𝑒 (𝑦) + 𝑒 𝑥 (𝑥 ) 𝑧𝑥 = 𝑒 𝑧𝑦 =

𝑥 𝑦

𝑦

−𝑒 𝑥𝑦 𝑦2

+

𝑦

+

𝑒𝑥 𝑥

( √𝑥 2 +𝑦 2 −𝑥

5.6) 𝑧 = 𝑙𝑛 (

√𝑥 2 +𝑦 2 +𝑥

)→

√𝑥2 +𝑦2 −𝑥 √𝑥2 +𝑦2 +𝑥

)

√𝑥2 +𝑦2 −𝑥 √𝑥2 +𝑦2 +𝑥

´

´

(√𝑥 2 + 𝑦 2 + 𝑥)(√𝑥 2 + 𝑦 2 − 𝑥) − (√𝑥 2 + 𝑦 2 + 𝑥) (√𝑥 2 + 𝑦 2 − 𝑥) 2

𝑧𝑦 =

(√𝑥 2 + 𝑦 2 + 𝑥) √𝑥 2 + 𝑦 2 − 𝑥 √𝑥 2 + 𝑦 2 + 𝑥

𝑧𝑦 =

𝑧𝑥 =

1 1 1 1 (√𝑥 2 𝑦 2 + 𝑥) [2 (𝑥 2 + 𝑦 2 )−2 . 2𝑦] − [2 (𝑥 2 + 𝑦 2 )−2 . 2𝑦] (√𝑥 2 𝑦 2 − 𝑥) 2

(√𝑥 2 + 𝑦 2 + 𝑥)

√𝑥 2 + 𝑦 2 − 𝑥 √𝑥 2 + 𝑦 2 + 𝑥 𝑦 𝑦 (√𝑥 2 𝑦 2 + 𝑥) ( )−( ) (√𝑥 2 + 𝑦 2 − 𝑥) 2 2 2 √𝑥 + 𝑦 √𝑥 + 𝑦 2

(√𝑥 2 𝑦 2 + 𝑥)(√𝑥 2 𝑦 2 − 𝑥) 𝑥𝑦 𝑥𝑦 (𝑦 + ) − (𝑦 − ) √𝑥 2 + 𝑦 2 √𝑥 2 + 𝑦 2 𝑧𝑥 = 𝑥2 + 𝑦2 − 𝑥2 𝑦√𝑥 2 + 𝑦 3 + 𝑥𝑦 − 𝑦√𝑥 2 + 𝑦 2 + 𝑥𝑦 √𝑥 2 𝑦 2 𝑧𝑥 = 𝑦2 1 2𝑥𝑦 2𝑥 𝑧𝑥 = = 2 2 2 𝑦 √𝑥 + 𝑦 𝑦√𝑥 2 + 𝑦 2

𝑦

5.7) 𝑧 = 𝑎𝑟𝑐𝑠𝑒𝑛 (𝑥 ) 𝑦 𝑥

( )

𝑧𝑥 =

√1−(𝑦)

→ 2´

𝑥

𝑧𝑥 =

𝑦 − 2 𝑥 2 2 √𝑥 −𝑦 𝑥2

→

𝑦 𝑥

( )

𝑧𝑦 =

√1−(𝑦)

𝑥(𝑦)´−𝑦(𝑥)´ 𝑥2 2 √1−𝑦2 𝑥 −𝑦 𝑥2

√𝑥2 −𝑦2

→ 2´

2

𝑥

𝑧𝑦 =

2 2 √𝑥 −𝑦 𝑥

→

𝑥2 𝑥 2 √𝑥 2 −𝑦 2

2

√1−𝑦2

−𝑦𝑥 𝑥 2 √𝑥 2 −𝑦 2

𝑥 𝑥(𝑦)´−𝑦(𝑥)´ 𝑥2

√1−𝑦2

− 2 𝑥 𝑥

→

−𝑦 𝑥√𝑥 2 −𝑦 2

𝑥

→

𝑥

𝑥

− 2 𝑥

→

𝑦

→

− 2 𝑥 2

√𝑥 2 −𝑦2 𝑥

=

𝑦

1 √𝑥 2 −𝑦 2

5.8) 𝑓(𝑥, 𝑦) = ∫𝑥 (𝑡 2 − 1)𝑑𝑡 𝑦 𝑡3 𝑥 3 −𝑦 3 𝑓(𝑥,𝑦) = [ 3 − 𝑡] → 𝑓(𝑐,𝑦) = ( 3 ) − (𝑥 − 𝑦) 𝑥 ′ 𝑥 3 −𝑦 3 ) 3 3𝑥 2 −1 3 2

𝑓(𝑥) = ( 𝑓(𝑥) =

𝑓(𝑥) = 𝑥 − 1

− (𝑥 − 𝑦)′

′ 𝑥 3 −𝑦3 ) − (𝑥 3 2 3𝑦 − − (−1) 3 2

𝑓(𝑦) = ( 𝑓(𝑦) =

− 𝑦)′

𝑓(𝑦) = 1 − 𝑦

5.10) 𝑔(𝑥,𝑦) = 𝑙𝑛√𝑥𝑦 𝑔𝑥 = 𝑔𝑥 =

(√𝑥𝑦)′ √𝑥𝑦

1 1 (𝑥𝑦)−2 .(𝑥𝑦)′ 2

√𝑥𝑦

𝑔(𝑥) = 𝑔(𝑥) = 𝑔(𝑦) = 𝑔(𝑦) =

1 (𝑥(𝑦)′ +𝑦(𝑥)′ ) 2√𝑥𝑦 𝑦 2√𝑥𝑦 √𝑥𝑦 1

√𝑥𝑦 𝑦

(√𝑥𝑦)′ √𝑥𝑦

𝑥 2√𝑥𝑦 √𝑥𝑦 1

1

= 2𝑥𝑦 → 2𝑥 →

=

1 ′ 1 (𝑥𝑦)−2(𝑥(𝑦)´+𝑦(𝑥) ) 2

√𝑥𝑦

𝑥 2𝑥𝑦

→

1 2𝑦

10. a) 𝑥 2 𝑦 − 𝑥𝑦 2 + 𝑦 2 = 7 𝑥 2 𝑦 − 𝑦 2 (𝑥 − 1) 𝑦(𝑥 2 − 𝑦(𝑥 − 1)) = 7 𝑦= 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥

=

7 (𝑥 2 −𝑦(𝑥−1)) ′

= =

2

(𝑥 2 −𝑦(𝑥−1)) −7(2𝑥−𝑦) 2

(𝑥 2 −𝑦(𝑥−1))

′

𝑑𝑦 𝑑𝑦

=

𝑑𝑦 𝑑𝑦

=

=

(𝑥 2 −𝑦(𝑥−1)).(7)′−7(𝑥 2 −𝑦(𝑥−1) (𝑥 2 −𝑦(𝑥−1)2 0−7(2𝑥−𝑦)

(𝑥 2 −𝑦(𝑥−1))(7)′(𝑥 2 −𝑦(𝑥−1)) 2

(𝑥 2 −𝑦(𝑥−1)) −7(0−(𝑥)) 2

(𝑥 2 −𝑦(𝑥−1)) 7𝑥 2

(𝑥 2 −𝑦(𝑥−1))

b)𝑦 3 + 𝑦 2 − 5𝑦 − 𝑥 2 + 4 = 0 𝑦 2 (𝑦 + 1) − 5𝑦 − 𝑥 2 + 4 = 0 𝑦[𝑦(𝑦 + 1) − 5] − 𝑥 2 + 4 = 0 −𝑦[𝑦(𝑦 + 1) − 5] + 𝑥 2 − 4 = 0

𝑥 2 = 4 + 𝑦[𝑦(𝑦 + 1) − 5] 𝑥 = √4 + 𝑦[𝑦(𝑦 + 1) − 5] 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥

1

1

1

1

= 2 (4 + 𝑦[𝑦(𝑦 + 1) − 5])−2 (4 + 𝑦[𝑦(𝑦 + 1) − 5])′ = 2 (4 + 𝑦[𝑦(𝑦 + 1) − 5])−2 [𝑦[𝑦(𝑦 + 1) − 5]′ + 𝑦[𝑦(𝑦 + 1) − 5](𝑦 ′ )]

𝑦[𝑦(𝑦 + 1)′ + (𝑦 + 1)(𝑦 ′ )][𝑦(𝑦 + 1) − 5] 𝑦[𝑦 + 𝑦 + 1] + [𝑦 2 + 𝑦 − 5] 2𝑦 2 + y + 𝑦 2 + y − 5 3𝑦 2 + 2𝑦 − 5 𝑑𝑦 𝑑𝑥

=

3𝑦 2 +2𝑦−5 2√4+𝑦[𝑦(𝑦+1)−5]

c) 𝑎𝑥 6 + 2𝑥 3 𝑦 − 𝑦 7 𝑥 = 10 𝑎𝑥 6 + 𝑥𝑦(2𝑥 2 − 𝑦 6 ) = 10 𝑎= 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥

=

𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥

𝑥6 ′ 𝑥 6 [10−𝑥𝑦(2𝑥 2 −𝑦6 )]′−[10−𝑥𝑦(2𝑥 2 −𝑦 6 )](𝑥 6 ) [𝑥 6 ]2 ′

= = = = = = = =

𝑎= 𝑑𝑎 𝑑𝑥 𝑑𝑎 𝑑𝑥

10−𝑥𝑦(2𝑥 2 −𝑦 6 )

𝑥 6 [0−𝑥𝑦(2𝑥 2 −𝑦6 ) +(2𝑥 2 −𝑦 6 )(𝑥𝑦)′]−[10−𝑥𝑦(2𝑥 2 −𝑦 6 )](6𝑥 5 ) 𝑥 12 𝑥 6 [−𝑥𝑦(4𝑥)+(2𝑥 2 −𝑦 6 )+(𝑦)]−[20𝑥 2 −10𝑦 6−2𝑥 3 𝑦+𝑥𝑦 7 ](6𝑥 5 ) 𝑥 12 𝑥 6 [−4𝑥 2 𝑦+2𝑥 2 𝑦−𝑦 7 ]−[120𝑥 7 −60𝑥 5 𝑦 6 −12𝑥 8 𝑦+6𝑥 6 𝑦 7 ] 𝑥 12 [−4𝑥 8 𝑦+2𝑥 8 𝑦−𝑥 6 𝑦 7 ]−[120𝑥 7 −60𝑥 5 𝑦 6 −12𝑥 8 𝑦+6𝑥 6 𝑦 7 ] 𝑥 12 [120𝑥 7 −60𝑥 5 𝑦 6 −12𝑥 8 𝑦+6𝑥 6 𝑦 7 ] 𝑥 12 [120𝑥 7 −60𝑥 5 𝑦 6 +10𝑥 8 𝑦−7𝑥 6 𝑦 7 ] 𝑥 12 𝑥 5 [120𝑥 2 +60𝑦 6 +10𝑥 8 𝑦+7𝑥𝑦 7 ] 𝑥 12 [120𝑥 7 −60𝑦6 −10𝑥 8 𝑦+7𝑥𝑦 7 ] 𝑥7

10−𝑥𝑦(𝑥 2 𝑦 6 𝑥6 ′ 𝑥 6 [10−𝑥𝑦(2𝑥 2 −𝑦 6 ] −[10−𝑥𝑦(2𝑥 2 −𝑦 6 ](𝑥 6)′

= = = = =

[𝑥 6 ]2 𝑥 6 [(𝑥𝑦)(2𝑥 2 −𝑦 6 )]′ 𝑥 12 ′ 𝑥 6 [(𝑥𝑦)(2𝑥 2 −𝑦 6 )] +(2𝑥 2 −𝑦6 (𝑥𝑦)′ ) 𝑥 12 −[(𝑥𝑦)[−6𝑦5 ]+(2𝑥 2 −𝑦 6 )(𝑥)] 𝑥6 −[−6𝑥6𝑦+2𝑥 3 +𝑥𝑦 6 ] 𝑥6

𝑥[6𝑥𝑦 6 −2𝑥 3 +𝑥𝑦 6 ] 𝑑𝑎 = 𝑑𝑥 𝑥6 𝑑𝑎 𝑥(6𝑦 6 −2𝑥 2 𝑦 6 = 𝑑𝑥 𝑥6 𝑑𝑎 6𝑦 6 −2𝑥 3 +𝑦 6 = 𝑑𝑥 𝑥5 𝑑𝑎 6𝑦 6 −2𝑥 2 +𝑦6 = 𝑑𝑥 𝑥5 𝑑𝑎 7𝑦 6 −2𝑥 2 = 𝑥5 𝑑𝑥

d) 𝑥 2 𝑦 = 4 + 2𝑥 − 3𝑦 2 − 5𝑦 3 𝑥 2 𝑦 = 4 + 2𝑥 − 𝑦 2 (3 − 5𝑦) 𝑦 2 (3 − 5𝑦) + 𝑥 2 𝑦 = 4 + 2𝑥 𝑦 2 (3 − 5𝑦) = 4 + 2𝑥 − 𝑥 2 𝑦 𝑦 2 (3 − 5𝑦) = 4 + 𝑥(2 − 𝑥𝑦) 𝑦2 =

4+𝑥(2−𝑥𝑦 3−5𝑦 4+𝑥(2−𝑥𝑦) 3−5𝑦

𝑦=√

1

1 4+𝑥(2−𝑥𝑦) −2 (3−5𝑦)(4+𝑥(2−𝑥𝑦)′ −(4+𝑥(2−𝑥𝑦))(3−5𝑦) ) ( ) (3−5𝑦)2 3−5𝑦

𝑑𝑦 𝑑𝑥

= 2(

𝑑𝑦 𝑑𝑥

= 2(

1

1 4+𝑥(2−𝑥𝑦) −2 (3−5𝑦)(𝑥(−𝑦)+2−𝑥𝑦) ) ( ) (3−5𝑦)2 3−5𝑦 1

𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥

→

1 4+𝑥(2−𝑥𝑦) −2 −2𝑥𝑦+2 ( 3−5𝑦 ) ( 3−5𝑦 ) 2 −2(𝑥𝑦−1)

= =

4+𝑥(2−𝑥𝑦) ) 3−5𝑦

2(3−5𝑦)(√ (𝑥𝑦−1)√3−5𝑦 √4+𝑥(2−𝑥𝑦)

4+𝑥(2−𝑥𝑦) 3−5𝑦

𝑦=√

1

=

1 4+𝑥(2−𝑥𝑦) −2 (3−5𝑦)(4+𝑥(2−𝑥𝑦)′ −(4+𝑥(2−𝑥𝑦)′ ( ) ( ) (3−5𝑦)2 2 3−5𝑦

=

1 4+𝑥(2−𝑥𝑦) −2 (3−5𝑦)(𝑥(−𝑥))−(4+𝑥(2−𝑥𝑦)(−5) ( ) ( ) (3−5𝑦)2 2 3−5𝑦

𝑑𝑦 𝑑𝑦

=

1 4+𝑥(2−𝑥𝑦) −2 −3𝑥 2 +5𝑥 2 𝑦+20+10𝑥−5𝑥 2 𝑦 ( 3−5𝑦 ) ( ) (3−5𝑦)2 2

𝑑𝑦 𝑑𝑦

1 4+𝑥(2−𝑥𝑦) −2 −3𝑥 2 +10𝑥+20 = ( ) ( (3−5𝑦)2 ) 2 3−5𝑦 2 −3𝑥 +10𝑥+20

𝑑𝑦 𝑑𝑦

1

𝑑𝑦 𝑑𝑦

1

1

=

4+𝑥(2−𝑥𝑦) (3−5𝑦)2 3−5𝑦 −3𝑥 2 +10𝑥+20

2√

=

4+𝑥(2−𝑥𝑦)√(3−5𝑦)3

11. 𝑓(𝑥,𝑦) = 𝐿𝑛(𝑆𝑒𝑛(𝑥 − 𝑦)) 𝑓𝑥 = 𝑓𝑥 = 𝑓𝑥𝑥 𝑓𝑥𝑥 𝑓𝑥𝑥 𝑓𝑥𝑥

(𝑆𝑒𝑛(𝑥−𝑦))

′

𝑆𝑒𝑛(𝑥−𝑦) 𝐶𝑜𝑠(𝑥−𝑦) = 𝑆𝑒𝑛(𝑥−𝑦)

𝐶𝑡𝑔(𝑥 − 𝑦)

= 𝐶𝑡𝑔(𝑥 − 𝑦) = −(𝑥 − 𝑦)′ . 𝐶𝑠𝑐 2 (𝑥 − 𝑦) = −(1)𝐶𝑠𝑐 2 (𝑥 − 𝑦) = −𝐶𝑠𝑐 2 (𝑥 − 𝑦)