Derivadas_parciales

Ap´
endice B

Lista complementaria de derivaci´
on
1. Calcula las derivadas parciales de las siguientes funciones:
(a) f(x, y, z) = x3y2 z − z 2y3 + 5xy2z − 2x5z 7
(b) h(p, q) =

p2 +q 2
2pq

(c) g(r, s, t) = (r2 + s2 + t2 )(r + s + t)
(d) l(x, y, w) = ln (x2y2 w) + ln (x + y + w3)
(e) g(r, s) = sin ( rs ) + cos ( r+s
)
r−s
(f) f(x, y) =

sin (x2 +y 2 )
cos (x2 +y 2 )

(g) h(m, n, p) = mn e(m
(h)f(x, y, z) = (x + y)z
2

(i) g(x, y, z, t) = e(x

2

+p2 )

3

+y 2 )

(z + t) + ln (xz) + yt cos (x + y)

sin (uv)
√
u2 +v 2 w 3

(j) l(u, v, w) =

(k) h(r, p, q) = esin (r

2

pq)

2

+ ln (cos (r + p + q))

ln (x +y +z 2 )
e(x+y+z)
2

(l) f(x, y, z) =

2

(m) g(w, t) = sin (wt) cos (w + t2) ln (w + t)
p2 + q2 + sin ( epq
pq )

(n) g(p, q) =

(o) h(m, n, p) =

3+mn
√
np
2 +y 2

(p) f(x, y, z,t) =

sin (x+y+z+t)+ex
cos (xyzt)

√
(q) g(u, v) = cos7 (u2 + v2 ) + ln3 ( u + v) + sin5 ( u+v
)
euv
√
u+v 5
2
2 7
3
(r) h(u, v) = cos (u + v ) + ln (( u + v) ) + sin ( euv )
(s) l(w, t, z) =
(t) f(r, s) =

5

7

(w + t2 + z 3 )4

r 2 +s2
cos (rs)

39

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´
APENDICE
B. LISTA COMPLEMENTARIA DE DERIVACION
1. Soluciones:
(a)
∂f
= 3x2y2 z + 5y2 z − 10x4z 7
∂x
∂f
= 2x3yz − 3z 2y2 + 10xyz
∂y
∂f
= x3 y2 −2zy3 + 5xy2 − 14x5z 6
∂z
(b)
∂h
p2 − q2
=
∂p
2p2q
∂h
q2 − p2
=
∂q
2pq2
(c)
∂g
= 3r2 + 2rs + 2rt + s2 + t2
∂r
∂g
= 3s2 + 2rs + 2st + r2 + t2
∂s
∂g
= 3t2 + 2ts + 2rt + s2 + r2
∂t
(d)
∂l
2
1
= +
∂x
x x + y + w3
∂l
2
1
= +
∂y
y
x + y + w3
∂l
1
3w2
= +
∂z
w x + y + w3
(e)
∂g
r+s
1
r
2s
sin (
= cos ( ) +
)
∂r
s
s
(r − s)2
r−s
∂g
r
r+s
−r
2s
sin (
= 2 cos ( ) −
)
2
∂s
s
s
(r − s)
r−s
(f)
∂f
2x(cos2 (x2+ y2 ) + sin2 (x2 + y2 ))
=
∂x
cos2 (x2 + y2 )
∂f
2y(cos2 (x2 + y2 ) + sin2 (x2 + y2 ))
=
∂y
cos2 (x2 + y2 )
(g)
2
2
∂h
= e(m +p ) (n + 2m2n)
∂m
2
2
∂h
= me(m +p )
∂n
2
2
∂h
= 2mnpe(m +p )
∂p

40

´
´
APENDICE
B. LISTA COMPLEMENTARIA DE DERIVACION

41

(h)
3
∂f
= z 3 (x + y)(z −1)
∂x
3
∂f
= z 3 (x + y)(z −1)
∂y
3
∂f
= (x + y)z 3z 2 ln (x + y)
∂z

(i)
2
2
∂g
1
= 2x(z + t)e(x +y ) + − yt sin (x +y)
∂x
x
2
2
∂g
= 2y(z + t)e(x +y ) + t cos (x + y) − yt sin (x + y)
∂y
2
2
∂g
1
= e(x +y ) +
∂z
z
2
2
∂g
= e(x +y ) + y cos (x + y)
∂t

(j)

√
u sin (uv)
v cos (uv) u2 + v2 w3 − √u2 +v2 w3
∂l
=
∂u
u2 + v 2 w 3
√
3
sin (uv)
√
u cos (uv) u2 + v2 w3 − vw
∂l
u2 +v 2 w 3
=
∂v
u2 + v 2 w 3
∂l
−3v2 w2 sin (uv)
√
=
∂w
2(u2 + v2 w3 ) u2 + v2 w3

(k)
2
∂h
− sin (r + p + q)
= esin (r pq) 2rpq cos (r2 pq) +∂r
cos (r + p + q)
2
∂h
− sin (r + p + q)
= esin (r pq) r2q cos (r2pq) +
∂p
cos (r + p + q)
2
∂h
− sin (r + p + q)
= esin (r pq) r2 p cos (r2 pq) +
∂q
cos (r + p + q)

(l)
∂f
=
∂x
∂f
=
∂y
∂f
=
∂z

2x
x2 +y 2 +z 2

− ln (x2 + y2 + z 2 )
e(x+y+z)

2y
x2 +y 2 +z 2

− ln (x2 + y2 + z 2 )
e(x+y+z)

2z
x2 +y 2 +z 2

− ln (x2 + y2 + z 2 )
e(x+y+z)

(m)
∂g
sin (wt) cos (w2 + t2)
= (t cos (wt) cos (w2 + t2 ) −2w sin (wt) sin (w2 + t2 )) ln (w + t) +
∂w
w+t
∂g
sin (wt) cos (w2 + t2 )
= (w cos (wt) cos (w2 + t2 ) − 2t sin (wt) sin (w2 + t2 )) ln (w + t) +
∂t
w+t

´
´
APENDICE
B. LISTA COMPLEMENTARIA DE DERIVACION

42

(n)
∂g
=
∂p
∂g
=
∂q

p
p2 + q2
q
p2 + q2

(o)

∂h
=
∂n

+

q − pq2
pq
cos ( pq )
epq
e

+

p − p2 q
pq
cos ( pq )
pq
e
e

√
np
∂h
=
∂m
p
√
√
m np − (3+mn)p
2 np
np

∂h
−(3 + mn)
=
√
∂p
2pnp
(p)
∂f
=
∂x
2

cos (xyzt) cos (x + y + z + t) + 2xex

+y 2

2

+ sin (x + y + z + t) + ex

+y 2

yzt sin (xyzt)

+y 2

xzt sin (xyzt)

cos2 (xyzt)
∂f
=
∂y
2

cos (xyzt) cos (x + y + z + t) + 2yex

+y 2

2

+ sin (x + y + z + t) + ex

cos2 (xyzt)
2

cos (xyzt) cos (x + y + z + t) + sin (x + y + z + t) + ex
∂f
=
∂z
cos2 (xyzt)

2

cos (xyzt) cos (x + y + z + t) + sin (x + y + z + t) + ex
∂f
=∂t
cos2 (xyzt)

+y 2

xyt sin (xyzt)

+y 2

xyz sin (xyzt)

(q)
√
∂g
3
= −14u cos6 (u2 + v2 ) sin (u2 + v2 ) +
ln2 ( u + v)+
∂u
2(u + v)
1 − uv − v2
u+v
u+v
sin4 ( uv ) cos ( uv )
uv
e
e
e
√
∂g
3
= −14v cos6 (u2 + v2 ) sin (u2 + v2 ) +
ln2 ( u + v)+
∂v
2(u + v)
+5

+5

1 − u2 − uv
u+v
u+v
sin4 ( uv ) cos ( uv )
euv
e
e

(r)
∂h
3/2
1 − uv − v2
= −14u(u2 + v2 )6 sin (u2 + v2 )7 +
+5
∂u
u+v
euv...