1)
lim
x
→
0
s
e
n
x
2
x
{\displaystyle \lim _{x\to 0}{\frac {senx}{2x}}}
1
2
lim
x
→
0
s
e
n
x
x
{\displaystyle {\frac {1}{2}}\lim _{x\to 0}{\frac {senx}{x}}}
1
2
.1
{\displaystyle {\frac {1}{2}}.1}
1
2
{\displaystyle {\frac {1}{2}}}
Jorgemontes 2)
lim
x
→
−
1
x
3
−
4
x
2
+
x
+
6
x
+
1
{\displaystyle \lim _{x\to -1}{\frac {x^{3}-4x^{2}+x+6}{x+1}}}
lim
x
→
−
1
(
x
+
1
)
(
x
2
−
5
x
+
6
)
x
+
1
{\displaystyle \lim _{x\to -1}{\frac {(x+1)(x^{2}-5x+6)}{x+1}}}
lim
x
→
−
1
x
2
−
5
x
+
6
{\displaystyle \lim _{x\to -1}\ {x^{2}-5x+6}}
(
−
1
)
2
−
5
(
−
1
)
+
6
{\displaystyle \ {(-1)^{2}-5(-1)+6}}
1
+
5
+
6
{\displaystyle \ {1+5+6}}
12
{\displaystyle \ {12}}
Jorgemontes 3)
lim
x
→
3
x
4
−
18
x
2
+
81
(
x
−
3
)
2
{\displaystyle \lim _{x\to 3}{\frac {x^{4}-18x^{2}+81}{(x-3)^{2}}}}
lim
x
→
3
(
x
−
3
)
(
x
3
+
3
x
2
−
9
x
−
27
)
(
x
−
3
)
2
{\displaystyle \lim _{x\to 3}{\frac {(x-3)(x^{3}+3x^{2}-9x-27)}{(x-3)^{2}}}}
lim
x
→
3
x
3
+
3
x
2
−
9
x
−
27
x
−
3
{\displaystyle \lim _{x\to 3}{\frac {x^{3}+3x^{2}-9x-27}{x-3}}}
lim
x
→
3
(
x
−
3
)
(
x
2
+
6
x
+
9
)
x
−
3
{\displaystyle \lim _{x\to 3}{\frac {(x-3)(x^{2}+6x+9)}{x-3}}}
lim
x
→
3
x
2
+
6
x
+
9
{\displaystyle \lim _{x\to 3}\ {x^{2}+6x+9}}
(
3
)
2
+
6
(
3
)
+
9
{\displaystyle \ {(3)^{2}+6(3)+9}}
9
+
18
+
9
{\displaystyle \ {9+18+9}}
36
{\displaystyle \ {36}}
Jorgemontes 4)
lim
h
→
0
(
x
+
h
)
2
−
x
2
h
{\displaystyle \lim _{h\to 0}{\frac {(x+h)^{2}-x^{2}}{h}}}
lim
h
→
0
x
2
+
2
h
x
+
h
2
−
x
2
h
{\displaystyle \lim _{h\to 0}{\frac {x^{2}+2hx+h^{2}-x^{2}}{h}}}
lim
h
→
0
2
h
x
+
h
2
h
{\displaystyle \lim _{h\to 0}{\frac {2hx+h^{2}}{h}}}
lim
h
→
0
(
h
)
(
2
x
+
h
)
h
{\displaystyle \lim _{h\to 0}{\frac {(h)(2x+h)}{h}}}
lim
h
→
0
(
2
x
+
h
)
{\displaystyle \lim _{h\to 0}\ {(2x+h)}}
(
2
x
)
{\displaystyle \ {(2x)}}
Jorgemontes 5)
lim
x
→
0
t
a
n
2
x
s
e
n
x
{\displaystyle \lim _{x\to 0}{\frac {tan^{2}x}{senx}}}
lim
x
→
0
(
s
e
n
2
x
/
c
o
s
2
x
)
s
e
n
x
{\displaystyle \lim _{x\to 0}{\frac {(sen^{2}x/cos^{2}x)}{senx}}}
lim
x
→
0
2
s
e
n
x
c
o
s
x
(
s
e
n
x
)
(
c
o
s
2
x
−
s
e
n
2
x
)
{\displaystyle \lim _{x\to 0}{\frac {2senxcosx}{(senx)(cos^{2}x-sen^{2}x)}}}
2
lim
x
→
0
c
o
s
x
(
c
o
s
2
x
−
s
e
n
2
x
)
{\displaystyle 2\lim _{x\to 0}{\frac {cosx}{(cos^{2}x-sen^{2}x)}}}
2
lim
x
→
0
c
o
s
x
c
o
s
2
x
−
(
1
−
c
o
s
2
x
)
{\displaystyle 2\lim _{x\to 0}{\frac {cosx}{cos^{2}x-(1-cos^{2}x)}}}
2
lim
x
→
0
c
o
s
x
2
c
o
s
2
x
−
1
{\displaystyle 2\lim _{x\to 0}{\frac {cosx}{2cos^{2}x-1}}}
2
lim
x
→
0
c
o
s
x
2
lim
x
→
0
2
c
o
s
2
x
−
1
{\displaystyle {\frac {2\lim _{x\to 0}\ cosx}{2\lim _{x\to 0}\ 2cos^{2}x-1}}}
(
2
)
(
1
)
(
2
)
(
1
)
−
1
{\displaystyle {\frac {(2)(1)}{(2)(1)-1}}}
2
{\displaystyle \ 2}
Jorgemontes 6)
1
3
x
2
{\displaystyle {\frac {1}{\sqrt[{2}]{3x}}}}
lim
h
→
0
1
3
x
+
3
h
2
−
1
3
x
2
h
{\displaystyle \lim _{h\to 0}{\frac {{\frac {1}{\sqrt[{2}]{3x+3h}}}-{\frac {1}{\sqrt[{2}]{3x}}}}{h}}}
lim
h
→
0
3
x
−
3
x
+
3
h
(
3
x
+
3
h
)
(
3
x
)
h
{\displaystyle \lim _{h\to 0}{\frac {\frac {{\sqrt {3x}}-{\sqrt {3x+3h}}}{({\sqrt {3x+3h}})({\sqrt {3x}})}}{h}}}
lim
h
→
0
3
x
−
3
x
+
3
h
(
h
)
(
(
3
x
+
3
h
)
(
3
x
)
)
{\displaystyle \lim _{h\to 0}{\frac {{\sqrt {3x}}-{\sqrt {3x+3h}}}{(h)(({\sqrt {3x+3h}})({\sqrt {3x}}))}}}
lim
h
→
0
3
x
−
3
x
+
3
h
(
h
)
(
(
3
x
+
3
h
)
(
3
x
)
)
{\displaystyle \lim _{h\to 0}{\frac {{\sqrt {3x}}-{\sqrt {3x+3h}}}{(h)(({\sqrt {3x+3h}})({\sqrt {3x}}))}}}
lim
h
→
0
3
x
−
3
x
+
3
h
(
h
2
)
(
(
3
x
+
3
h
)
(
3
x
)
)
{\displaystyle \lim _{h\to 0}{\frac {{\sqrt {3x}}-{\sqrt {3x+3h}}}{({\sqrt {h^{2}}})(({\sqrt {3x+3h}})({\sqrt {3x}}))}}}
lim
h
→
0
3
x
−
3
x
+
3
h
(
(
h
2
)
(
3
x
2
+
9
h
x
)
)
{\displaystyle \lim _{h\to 0}{\frac {{\sqrt {3x}}-{\sqrt {3x+3h}}}{({\sqrt {(h^{2})(3x^{2}+9hx)}})}}}
lim
h
→
0
3
x
−
3
x
+
3
h
(
3
x
2
h
2
+
9
h
3
x
)
{\displaystyle \lim _{h\to 0}{\frac {{\sqrt {3x}}-{\sqrt {3x+3h}}}{\sqrt {(3x^{2}h^{2}+9h^{3}x)}}}}
lim
h
→
0
3
x
−
3
x
+
3
h
3
x
2
h
2
+
9
h
3
x
{\displaystyle \lim _{h\to 0}{\frac {{\sqrt {3x}}-{\sqrt {3x+3h}}}{\sqrt {3x^{2}h^{2}+9h^{3}x}}}}
lim
h
→
0
3
x
−
3
x
+
3
h
(
3
h
2
x
)
(
3
x
+
3
h
)
{\displaystyle \lim _{h\to 0}{\frac {{\sqrt {3x}}-{\sqrt {3x+3h}}}{\sqrt {(3h^{2}x)(3x+3h)}}}}
1)
lim
h
→
0
(
2
+
h
)
2
−
4
h
{\displaystyle \lim _{h\to 0}{\frac {(2+h)^{2}-4}{h}}}
lim
h
→
0
4
+
4
h
+
h
2
−
4
h
{\displaystyle \lim _{h\to 0}{\frac {4+4h+h^{2}-4}{h}}}
lim
h
→
0
4
h
+
h
2
h
{\displaystyle \lim _{h\to 0}{\frac {4h+h^{2}}{h}}}
lim
h
→
0
h
(
4
+
h
)
h
{\displaystyle \lim _{h\to 0}{\frac {h(4+h)}{h}}}
lim
h
→
0
4
+
h
{\displaystyle \lim _{h\to 0}\ {4+h}}
4
+
0
{\displaystyle \ {4+0}}
4
{\displaystyle \ {4}}
Jorgemontes
2)Derivar:
F
(
x
)
=
(
x
+
1
)
2
x
−
1
{\displaystyle \ F(x)={\frac {(x+1)^{2}}{x-1}}}
lim
h
→
0
(
(
(
x
+
h
)
+
1
)
2
)
(
x
+
h
)
−
1
−
(
x
+
1
)
2
x
−
1
h
{\displaystyle \lim _{h\to 0}{\frac {{\frac {(((x+h)+1)^{2})}{(x+h)-1}}-{\frac {(x+1)^{2}}{x-1}}}{h}}}
lim
h
→
0
h
x
2
+
h
2
x
−
3
h
−
2
h
x
(
x
+
h
−
1
)
(
x
−
1
)
h
{\displaystyle \lim _{h\to 0}{\frac {\frac {hx^{2}+h^{2}x-3h-2hx}{(x+h-1)(x-1)}}{h}}}
lim
h
→
0
(
h
)
(
x
2
+
h
x
−
3
−
2
x
)
(
h
)
(
(
x
+
h
−
1
)
(
x
−
1
)
)
{\displaystyle \lim _{h\to 0}{\frac {(h)(x^{2}+hx-3-2x)}{(h)((x+h-1)(x-1))}}}
lim
h
→
0
x
2
+
h
x
−
3
−
2
x
(
x
+
h
−
1
)
(
x
−
1
)
{\displaystyle \lim _{h\to 0}{\frac {x^{2}+hx-3-2x}{(x+h-1)(x-1)}}}
x
2
−
2
x
−
3
(
x
−
1
)
2
{\displaystyle {\frac {x^{2}-2x-3}{(x-1)^{2}}}}
Jorgemontes 3)Hallar la derivada de:
y=x
y
′
=
1
{\displaystyle \ y'=1}
l
i
m
x
→
0
f
(
x
+
h
)
−
F
(
x
)
h
{\displaystyle \ lim_{x\to 0}{\frac {f(x+h)-F(x)}{h}}}
l
i
m
x
→
0
x
+
h
−
x
h
{\displaystyle \ lim_{x\to 0}{\frac {x+h-x}{h}}}
l
i
m
x
→
0
h
h
{\displaystyle \ lim_{x\to 0}{\frac {h}{h}}}
1
{\displaystyle \ 1}
by: jorge montes and Zorraidorsito Jorgemontes
4)Derive:
6
x
2
+
1
{\displaystyle {\frac {6}{x^{2}+1}}}
lim
h
→
0
6
(
x
+
h
)
2
+
1
−
6
x
2
+
1
h
{\displaystyle \lim _{h\to 0}{\frac {{\frac {6}{(x+h)^{2}+1}}-{\frac {6}{x^{2}+1}}}{h}}}
lim
h
→
0
6
x
2
+
6
−
6
x
2
−
12
h
x
−
6
h
2
−
6
(
(
x
+
h
)
2
+
1
)
(
x
2
+
1
)
h
{\displaystyle \lim _{h\to 0}{\frac {\frac {6x^{2}+6-6x^{2}-12hx-6h^{2}-6}{((x+h)^{2}+1)(x^{2}+1)}}{h}}}
lim
h
→
0
−
6
h
2
−
12
h
x
(
(
x
+
h
)
2
+
1
)
(
x
2
+
1
)
h
{\displaystyle \lim _{h\to 0}{\frac {\frac {-6h^{2}-12hx}{((x+h)^{2}+1)(x^{2}+1)}}{h}}}
lim
h
→
0
−
6
h
2
−
12
h
x
(
h
)
(
(
(
x
+
h
)
2
+
1
)
(
x
2
+
1
)
)
{\displaystyle \lim _{h\to 0}{\frac {-6h^{2}-12hx}{(h)(((x+h)^{2}+1)(x^{2}+1))}}}
lim
h
→
0
(
h
)
(
−
6
h
−
12
x
)
(
h
)
(
(
(
x
+
h
)
2
+
1
)
(
x
2
+
1
)
)
{\displaystyle \lim _{h\to 0}{\frac {(h)(-6h-12x)}{(h)(((x+h)^{2}+1)(x^{2}+1))}}}
lim
h
→
0
−
6
h
−
12
x
(
(
x
+
h
)
2
+
1
)
(
x
2
+
1
)
{\displaystyle \lim _{h\to 0}{\frac {-6h-12x}{((x+h)^{2}+1)(x^{2}+1)}}}
−
6
(
0
)
−
12
x
(
(
x
+
(
0
)
)
2
+
1
)
(
x
2
+
1
)
{\displaystyle {\frac {-6(0)-12x}{((x+(0))^{2}+1)(x^{2}+1)}}}
−
12
x
(
x
2
+
1
)
2
{\displaystyle {\frac {-12x}{(x^{2}+1)^{2}}}}
Jorgemontes 5)Derive por implícita:
y
2
−
x
2
=
1
{\displaystyle \ {y^{2}-x^{2}=1}}
d
d
x
y
2
−
d
d
x
x
2
=
d
d
x
1
{\displaystyle \ {{\frac {d}{dx}}y^{2}-{\frac {d}{dx}}x^{2}={\frac {d}{dx}}1}}
d
d
x
2
y
−
d
d
x
2
x
=
d
d
x
0
{\displaystyle \ {{\frac {d}{dx}}2y-{\frac {d}{dx}}2x={\frac {d}{dx}}0}}
d
d
x
2
y
−
d
d
x
2
x
=
0
{\displaystyle \ {{\frac {d}{dx}}2y-{\frac {d}{dx}}2x=0}}
d
d
x
(
2
y
−
2
x
)
=
0
{\displaystyle \ {{\frac {d}{dx}}(2y-2x)=0}}
d
d
x
(
2
y
)
=
2
x
{\displaystyle \ {{\frac {d}{dx}}(2y)=2x}}
d
d
x
=
2
x
2
y
{\displaystyle \ {{\frac {d}{dx}}={\frac {2x}{2y}}}}
d
d
x
=
x
y
{\displaystyle \ {{\frac {d}{dx}}={\frac {x}{y}}}}
6)calcular
f
,
(
0
)
,
f
,
(
1
2
)
,
f
,
(
1
)
,
f
,
(
−
10
)
{\displaystyle \ f^{,}(0),f^{,}({\frac {1}{2}}),f^{,}(1),f^{,}(-10)}
f
(
x
)
=
2
+
x
−
x
2
{\displaystyle \ {f(x)=2+x-x^{2}}}
f
,
(
x
)
=
1
−
2
x
{\displaystyle \ {f^{,}(x)=1-2x}}
f
,
(
0
)
=
1
{\displaystyle \ {f^{,}(0)=1}}
f
,
(
1
2
)
=
0
{\displaystyle \ {f^{,}({\frac {1}{2}})=0}}
f
,
(
1
)
=
−
1
{\displaystyle \ {f^{,}(1)=-1}}
f
,
(
−
10
)
=
−
19
{\displaystyle \ {f^{,}(-10)=-19}}
Encuentre la ecuación de la recta tangente en el punto propuesto
1)
x
3
y
+
y
3
x
=
30
{\displaystyle \ {x^{3}y+y^{3}x=30}}
1)Se da una tabla de valores para f, g, f´ y g´
/ x / f(x) / g(x) / f´(x) / g´(x) / / 1 / 3 / 2 / 4 / 6 /
/ 2 / 1 / 8 / 5 / 7 /
/ 3 / 7 / 2 / 7 / 9 / Si h(x) = g(f(x)) , el valor de h´(1) es igual a:
a)4 b)5 c)36 d)24 e)30 R/ 2)
3)Un rectángulo tiene dos vértices sobre el eje x y otros dos sobre la parabola
y
=
12
−
x
2
{\displaystyle \ {y=12-x^{2}}}
A
=
(
2
x
)
(
y
)
{\displaystyle \ {A=(2x)(y)}}
A
=
(
2
x
)
(
12
−
x
2
)
{\displaystyle \ {A=(2x)(12-x^{2})}}
A
=
24
x
−
2
x
3
{\displaystyle \ {A=24x-2x^{3}}}
A
,
=
24
−
6
x
2
{\displaystyle \ {A^{,}=24-6x^{2}}}
0
=
24
−
6
x
2
{\displaystyle \ {0=24-6x^{2}}}
24
=
6
x
2
{\displaystyle \ {24=6x^{2}}}
24
6
=
x
2
{\displaystyle \ {{\frac {24}{6}}=x^{2}}}
4
=
x
2
{\displaystyle \ {4=x^{2}}}
4
=
x
{\displaystyle \ {{\sqrt {4}}=x}}
2
=
x
{\displaystyle \ {2=x}}
y
=
12
−
x
2
{\displaystyle \ {y=12-x^{2}}}
y
=
12
−
(
2
)
2
{\displaystyle \ {y=12-(2)^{2}}}
y
=
12
−
4
{\displaystyle \ {y=12-4}}
y
=
8
{\displaystyle \ {y=8}}
A
=
(
2
(
2
)
)
(
8
)
{\displaystyle \ {A=(2(2))(8)}}
A
=
(
4
)
(
8
)
{\displaystyle \ {A=(4)(8)}}
A
=
32
{\displaystyle \ {A=32}}
--Jorgemontes 13:07, 19 Jul 2004 (UTC)
4)
![{\displaystyle {\frac {1}{\sqrt[{2}]{3x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7baf93bbda6139f64ecb83069db088a452516a06)
![{\displaystyle \lim _{h\to 0}{\frac {{\frac {1}{\sqrt[{2}]{3x+3h}}}-{\frac {1}{\sqrt[{2}]{3x}}}}{h}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33c480795fc1b0328e3b9dfd940b49f1c736cb12)
