Lista_3_Mat2_2013-2_T02

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UFRRJ - ICE - DEMAT
Prof.a Aline
3a Lista de Exerc´ıcios de Matema´tica II - 2013-2
Calcule as derivadas parciais fx e fy de cada func¸a˜o dada a seguir:
(a) f(x, y) = 2xy5 + 3x2y + x2
(b) f(x, y) = (3x + 2y)5
(c) f(x, y) =
3y
2x
(d) f(x, y) = xexy
(e) f(x, y) =
e2−x
y2
(f) f(x, y) =
2x + 3y
y − x
(g) f(x, y) = x ln y
(h) f(x, y) =
ln(x + 2y)
y2
(i) f(x, y) = 2x3y + 3xy2 +
y
x
(j) f(x, y) = xyexy
(k) f(x, y) = ln
(
xy
x + 3y
)
Gabarito:
(a) fx = 2y
5 + 6xy + 2x, fy = 10xy
4 + 3x2; (b) fx = 15(3x + 2y)
4, fy = 10(3x + 2y)
4;
(c) fx = −
3y
2x2
, fy =
3
2x
; (d) fx = (xy + 1)e
xy, fy = x
2exy; (e) fx =
−e2−x
y2
, fy =
−2e2−x
y3
;
(f) fx =
5y
(y − x)2
, fy =
−5x
(y − x)2
; (g) fx = ln y, fy =
x
y
;
(h) fx =
1
y2(x + 2y)
, fy =
2[y − (x + 2y) ln(x + 2y)]
y3(x + 2y)
;
(i) fx = 6x
2y + 3y2 −
y
x2
, fy = 2x
3 + 6xy +
1
x
; (j) fx = y(xy + 1)e
xy, fy = x(xy + 1)e
xy;
(k) fx =
3y
x(x + 3y)
=
1
x
−
1
x + 3y
, fy =
x
y(x + 3y)
=
1
y
−
3
x + 3y
·