Find f x and f y. f ( x, y ) = x 5 + y 5 + x 5y f x = 5x 4 + 5x 4y, f y = 5y 4 + x 5 f x = 2x + 2xy, f y = 2y + x 2 f x = 3x 2 + 3x 2y, f y = 3y 2 + x 3 f x = 4x 3 + 4x 3y, f y = 4y 3 + x 4 1. 2. 3. 4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Find the indicated partial derivatives Find the indicated partial derivatives. f ( x, y ) = sin ( 4x + 2y ) ; f y ( - 4, 8 ) f y ( - 4, 8 ) = - 9 f y ( - 4, 8 ) = 5 f y ( - 4, 8 ) = 2 f y ( - 4, 8 ) = - 8 1. 2. 3. 4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Find all the second partial derivatives. f ( x, y ) = x 4 - 9x 2y 3 1. fxx = 2x 2 - 54y 3, fxy = - 54xy 2, fyx = - 54xy 2, fyy = - 54x 2y fxx = 12x 2 - 18y 3, fxy = - 54xy 2, fyx = - 54xy 2, fyy = - 54x 2y fxx = 2x 2 - 18y 3, fxy = - 54xy 2, fyx = - 54xy 2, fyy = - 54x 2y fxx = - 12x 2 + 18y 3, fxy = 54xy 2, fyx = 54xy 2, fyy = 54x 2y 2. 3. 4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

If k, l, m are the sides of a triangle and K, L, M are the opposite angles, find {image} by implicit differentiation of the Law of Cosines. 1. {image} 2. 3. 4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50