Ejercisios de calculo

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| 2011 |
| Cea 08 Pueblo Nuevo Oax.

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[EJERCICIOS DE CALCULO DIFERENCIAL] |
Este trabajo contiene ejercicios del libro Calculo Direncial E Integral GRANVILLE, pág. 44-45 ejercicios del 9-36 resueltos paso a paso. |

Ejercicios de Calculo
Comprobar cada una de las siguientes derivadas.
1.- d/dx (3x4 – 2x2 + 8) =
= d/dx(3x4) - d/dx (2x2) + d/dx (8)
= 3d/dx (x4) – 2d/dx (x2)+d/dx (8)
= 3.4 x3 d/dx (x) – 2.2x d/dx(x)
= 12x3 – 4x

2.- d/dx (4 + 3x – 2x3)
= d/dx (4) + d/dx (3x) – d/dx(2x3)
= d/dx (4) + 3d/dx (x) – 2 d/dx(x3)
= 3 – 2.3x2 d/dx (x)
= 3 – 6x2

3.- d/dt (at5 – 5bt3) =
= d/dt (at5) – d/dt (5bt3)
= a d/dt (t5) – 5b d/dt (t3)
= a.5 t4 d/dt (t) – 5b.3 t2 d/dt (t)
= 5at4 – 15bt2

4.- d/dz (z2/2 – z7/7) =
= d/dz (z2/2) – d/dz (z7/7)
= ½ d/dz(z2) – 1/7 d/dz (z7)
= ½ .2 d/dz (z) – 1/7 .7 z6 d/dz (z)
= 2/2 z – 7/7 z6
= z – z6

5.- d/dx √v
= d/dx v1/2
= ½ v ½-1 dv/dx = = 1/2√v dv/dx
= ½ v-1/2 dv/dx
= 1/2√v dv/dx

6.- d/dx (2/x – 3/x2)
= d/dx (2/x) – d/dx (3/x2)
= 2 d/dx (1/x) – 3 d/dx (1/x2)
= 2 d/dx (x-1) – 3 d/dx (x-2)
= 2(-1)x-1-1 d/dx (x) – 3(-2)x-2-1 d/dx (x)
= -2x-2 + 6x-3 = -2/x2 + 6/x3

7.- d/dt (2t4/3 – 3t2/3)= d/dt (2t4/3) – d/dt (3t2/3)
= 2d/dt (t4/3) – 3d/dt (t2/3)
= 2(4/3)t4/3-1 d/dt (t) – 3 (2/3)t2/3-1 d/dt (t)
= 8/3t1/3 – 2t-1/3

8.- d/dx (2x3/4 + 4x-1/4)
= d/dx (2x3/4) + d/dx (4x-1/4)
=2d/dx (x3/4) + 4 d/dx(x-1/4)
= 2(3/4) x3/4-1 d/dx (x) + 4(-1/4) x-1/4-1 d/dx (x)
= 3/2 x -1/4 – x-5/4

9.- d/dx (x2/3 – a2/3)
= d/dx (x2/3) – d/dx (a2/3)
= 2/3x2/3-1 d/dx (x)
= 2/3x-1/3

10.-d/dx (a + bx + cx2/ x)
= d/dx (a/x + bx/x + cx2/x)
= d/dx (a/x + b + cx)
= d/dx (a/x) + d/dx (b) + d/dx (cx)
= a d/dx (1/x) + c d/dx (x)
= a d/dx (x-1) + c
= a (-1) x-1-1 d/dx (x) + c
= -ax-2 + c
= c- a/x2

11.- d/dx(√x/2 – 2/√x)
=d/dx (√x/2) – (2/√x)
= 1/2 d/dx (√x) – 2 d/dx (1/√x)
= ½ d/dx (x1/2) – 2 d/dx (x-1/2)
= ½ (1/2) x1/2-1 d/dx (x) – 2 (-1/2) x-1/2-1 d/dx (x)
= ¼ x1/2 +x3/2
= 1/4√x + 1/x3/2
= ¼√x + 1/√x3
= ¼√x + 1√x2.x
= 1/4√x + 1/x√x

12.- d/dt (a + bt + ct2 / √t)
= d/dt (a/t + bt/√t + ct2/√t)
= d/dt (t1/2a + t1/2bt + t1/2ct2)
= d/dt (at1/2 + bt1/2 + ct3/2)
= d/dt (at1/2) + d/dt (bt1/2) + d/dt (ct3/2)
= a d/dt (t1/2) + b d/dt (t1/2) + c d/dt (t3/2)
= a (-1/2) t-1/2 -1 d/dt (t) + b(1/2)t1/2 – 1 d/dt (t) + c(3/2) t3/2 -1 d/dt (t)
= -a/2 t3/2 + b/2t1/2 + 3/2 c t1/2
= -a/2√t3 + b/2√t + 3c√t/2
= -a/2√t2.t + b/2√t + 3c√t/2
= -a/2t√t + b/2√t + 3c√t/2

13.- d/dx (√ax + a/√ax)
= d/dx (√ax) + d/dx (a/√ax)
= d/dx (ax)1/2 + a d/dx (1/√ax)
= d/dx (a1/2x1/2) + a d/dx (ax)1/2
= a1/2 d/dx (x1/2) + a d/dx (a-1/2 x-1/2)
= √a d/dx(x1/2) + a.a1/2 d/dx (x-1/2)
= √a d/dx (x1/2) x a1/2 d/dx (x-1/2)
=√a d/dx (x1/2) + √a d/dx (x-1/2)
= √a (½) x1/2-1 d/dx (x) + √a (-½) x-1/2 -1 d/dx (x)
= √a/2 x-1/2 - √a/2 x-3/2 = √a/2√x - √a/2x3
= √a/2√x - √a/2√x2.x = √a/2√x - √a/2x√x
14.- d/dᶿ √1 – 2ᶿ
= d/dᶿ (1 – 2ᶿ)1/2
= ½ (1 – 2ᶿ)1/2 -1 d/dᶿ (1 – 2ᶿ)
= ½ (1 – 2ᶿ)-1/2 [d(1) – d(2ᶿ)]
= 1/2√1 - 2ᶿ [d(1) – 2 dᶿ/dᶿ]
= -2/2√1 - 2ᶿ
= 1/√1 - 2ᶿ

15.- d/dt (2 – 3t2)
= 3(2 – 3t2)3-1 d/dt (2 – 3t2)
= 3(2 – 3t2) [d/dt (2) – d/dt (3t2)]
= 3(2 – 3t2)2[-3.2t d/dt (t)]
= 3(2 – 3t2)2 (-6t)
= -18t (2 – 3t2)2

16.- d/dx 3√4-9x
= d/dx (4 – 9x)1/3
= 1/3 (4 – 9x)1/3-1 d/dx(4 – 9x)
= 1/3 (4 – 9x)-2/3 [d(dx(4) – d/dx(9)]
=1/3 (4 – 9x)-2/3 [-9 d/dx (x)]
= -9/3 (4 – 9x)-2/3
= -3 (4 – 9x) -2/3
= -3/(4 – 9x)-2/3
17.- d/dx (1/√a2+x2)
= d/dx (a2+x2)-1/2
= -½ (a2+x2)-1/2-1 d/dx (a2+x2)
= -½ (a2+x2)-3/2 [d/dx (a2) – d/dx (x2)]
= -½(a2+x2)-3/2[2x d/dx (x)]
= 2x/2 (a2+x2)-3/2
= x/( a2+x2)3/2

18.- d/dx (a – b/x)2
= 2(a-b/x)2-1 d/dx (a- b/x)
= 2 (a-b/x)1 [d/dx (a) – d/dx (b/x)]
= 2 (a-b/x) [-b d/dx (x-1)
= 2 (a-b/x) [-b (-1) x-1-1 d/dx (x)]
= 2 (a-b/x) (bx-2)
= 2b/x2 (a-b/x)

19.- d/dx (a + b/x2)3
= 3 (a + b/x2)3-1 d/dx (a + b/x2)
= 3 (a + b/x2)2 [d/dx (a) + d/dx (b/x2]
= 3 (a + b/x2)2 [b d/dx (x-2)]
= 3 (a + b/x2)2 [b...
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