Tablas De Ayuda De Derivadas.pdf
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ΓLGEBRA DE DERIVADAS
π π π (π(π₯) Β± π(π₯)) = π(π₯) Β± π(π₯) ππ₯ ππ₯ ππ₯
π ππ(π₯) ππ₯
π π π (π(π₯)π(π₯)) = π(π₯) ( π(π₯)) + ( π(π₯)) π(π₯) ππ₯ ππ₯ ππ₯
π π(π₯) ( )= ππ₯ π(π₯)
=π
π π(π₯); ππ₯
π(π₯) (
π: πΆπππ π‘πππ‘π
π π π(π₯)) β π(π₯) ( π(π₯)) ππ₯ ππ₯ [π(π₯)]2
π π(π₯) π π (π(π₯)π(π₯) ) = π(π₯)π(π₯) [ ( π(π₯)) + πΏπ(π(π₯)) ( π(π₯))] ππ₯ π(π₯) ππ₯ ππ₯ TABLA DE DERIVADAS
NΒΊ
FUNCIΓN
DERIVADA
1
π¦ = π; π: πΆπππ π‘πππ‘π
2
π¦=π₯
3
π¦ = π₯π
ππ =0 ππ₯ ππ₯ =1 ππ₯ π π ππ₯ π₯ = ππ₯ πβ1 ππ₯ ππ₯
4
π¦ = ππππ (π₯)
5
π¦ = πΏπ(π₯)
6
π¦ = ππ₯
7
π¦ = ππ₯
8
π¦ = π ππ(π₯)
9
π¦ = πππ (π₯)
10
π¦ = π‘ππ(π₯)
11
π¦ = πππ‘(π₯)
12
π¦ = π ππ(π₯)
13
π¦ = ππ π(π₯)
14
π¦ = ππππ ππ(π₯)
15
π¦ = ππππππ (π₯)
16
π¦ = ππππ‘ππ(π₯)
17
π¦ = π ππβ(π₯)
18
π¦ = πππ β(π₯)
19
π¦ = π‘ππβ(π₯)
20
π¦ = ππππ ππβ(π₯)
21
π¦ = ππππππ β(π₯)
22
π¦ = ππππ‘ππβ(π₯)
π ππ₯
DERIVADA COMPUESTA
1
ππ₯
ππππ (π₯)= π₯ ππππ (π) ππ₯
π 1 ππ₯ πΏπ(π₯) = ππ₯ π₯ ππ₯ π π₯ ππ₯ π = π π₯ πΏπ(π) ππ₯ ππ₯ π π₯ ππ₯ π = ππ₯ ππ₯ ππ₯ π ππ₯
ππ₯
π ππ(π₯ ) = cos(π₯) ππ₯
π ππ₯ πππ (π₯ ) = βπ ππ(π₯) ππ₯ ππ₯ π ππ₯
1
ππ₯
π‘ππ(π₯ ) = πππ 2π₯ ππ₯
π ππ₯ πππ‘(π₯ ) = βππ π 2 (π₯) ππ₯ ππ₯ π ππ₯ π ππ (π₯ ) = sec(π₯) tan(π₯) ππ₯ ππ₯ π ππ₯ ππ π (π₯ ) = βππ π(π₯)πππ‘(π₯) ππ₯ ππ₯ π 1 ππ₯ ππππ ππ (π₯) = ππ₯ β1 β π₯ 2 ππ₯ π β1 ππ₯ ππππππ (π₯ ) = ππ₯ β1 β π₯ 2 ππ₯ π ππ₯
1
ππ₯
ππππ‘ππ(π₯ ) = 1+π₯ 2 ππ₯
π ππ₯ π ππβ(π₯) = πππ β(π₯) ππ₯ ππ₯ π ππ₯ πππ β(π₯ ) = π ππβ(π₯ ) ππ₯ ππ₯ π 1 ππ₯ π‘ππβ(π₯ ) = 2 ππ₯ πππ β (π₯) ππ₯ π 1 ππ₯ ππππ ππβ(π₯ ) = ππ₯ βπ₯ 2 + 1 ππ₯ π 1 ππ₯ ππππππ β(π₯ ) = 2 ππ₯ βπ₯ β 1 ππ₯ π 1 ππ₯ ππππ‘ππβ(π₯ ) = ππ₯ 1 β π₯ 2 ππ₯
π π (π (π₯ ))π = ππ (π₯ )πβ1 π(π₯) ππ₯ ππ₯ π 1 π ππππ (π(π₯ ))= π(π₯) ππππ (π) ππ₯ π(π₯) ππ₯ π 1 π πΏπ(π (π₯ )) = π(π₯) ππ₯ π (π₯ ) ππ₯ π π(π₯) π π = π π(π₯) πΏπ(π) π (π₯ ) ππ₯ ππ₯ π π(π₯) π π = π π(π₯) π(π₯) ππ₯ ππ₯ π π π ππ(π (π₯ )) = cos(π(π₯ )) π (π₯ ) ππ₯ ππ₯ π π πππ (π (π₯)) = βπ ππ(π(π₯ )) π (π₯ ) ππ₯ ππ₯ π 1 π π‘ππ(π (π₯ )) = π (π₯ ) ππ₯ πππ 2 (π(π₯ )) ππ₯ π π πππ‘(π (π₯ )) = βππ π 2 (π (π₯)) π (π₯ ) ππ₯ ππ₯ π π π ππ(π (π₯ )) = sec(π (π₯ )) tan(π(π₯ )) π(π₯ ) ππ₯ ππ₯ π π ππ π(π (π₯)) = βππ π(π(π₯ ))πππ‘(π(π₯ )) π (π₯ ) ππ₯ ππ₯ π 1 π ππππ ππ(π (π₯ )) = π (π₯ ) ππ₯ β1 β (π(π₯ ))2 ππ₯ π β1 π ππππππ (π (π₯ )) = π (π₯ ) 2 ππ₯ ππ₯ β1 β (π (π₯)) π 1 π ππππ‘ππ(π (π₯)) = π (π₯ ) ππ₯ 1 + (π(π₯ ))2 ππ₯ π π π ππβ(π (π₯ )) = πππ β(π(π₯ )) π(π₯ ) ππ₯ ππ₯ π π πππ β(π(π₯ )) = π ππβ(π (π₯ )) π (π₯ ) ππ₯ ππ₯ π 1 π π‘ππβ(π (π₯ )) = π (π₯ ) 2 ππ₯ πππ β (π(π₯ )) ππ₯ π 1 π ππππ ππβ(π (π₯ )) = π (π₯ ) ππ₯ β(π (π₯ ))2 + 1 ππ₯ π 1 π ππππππ β(π (π₯ )) = π (π₯ ) ππ₯ β(π(π₯ ))2 β 1 ππ₯ π 1 π ππππ‘ππβ(π (π₯ )) = π (π₯ ) 2 ππ₯ 1 β (π(π₯)) ππ₯
π π π (π(π₯) Β± π(π₯)) = π(π₯) Β± π(π₯) ππ₯ ππ₯ ππ₯
π ππ(π₯) ππ₯
π π π (π(π₯)π(π₯)) = π(π₯) ( π(π₯)) + ( π(π₯)) π(π₯) ππ₯ ππ₯ ππ₯
π π(π₯) ( )= ππ₯ π(π₯)
=π
π π(π₯); ππ₯
π(π₯) (
π: πΆπππ π‘πππ‘π
π π π(π₯)) β π(π₯) ( π(π₯)) ππ₯ ππ₯ [π(π₯)]2
π π(π₯) π π (π(π₯)π(π₯) ) = π(π₯)π(π₯) [ ( π(π₯)) + πΏπ(π(π₯)) ( π(π₯))] ππ₯ π(π₯) ππ₯ ππ₯ TABLA DE DERIVADAS
NΒΊ
FUNCIΓN
DERIVADA
1
π¦ = π; π: πΆπππ π‘πππ‘π
2
π¦=π₯
3
π¦ = π₯π
ππ =0 ππ₯ ππ₯ =1 ππ₯ π π ππ₯ π₯ = ππ₯ πβ1 ππ₯ ππ₯
4
π¦ = ππππ (π₯)
5
π¦ = πΏπ(π₯)
6
π¦ = ππ₯
7
π¦ = ππ₯
8
π¦ = π ππ(π₯)
9
π¦ = πππ (π₯)
10
π¦ = π‘ππ(π₯)
11
π¦ = πππ‘(π₯)
12
π¦ = π ππ(π₯)
13
π¦ = ππ π(π₯)
14
π¦ = ππππ ππ(π₯)
15
π¦ = ππππππ (π₯)
16
π¦ = ππππ‘ππ(π₯)
17
π¦ = π ππβ(π₯)
18
π¦ = πππ β(π₯)
19
π¦ = π‘ππβ(π₯)
20
π¦ = ππππ ππβ(π₯)
21
π¦ = ππππππ β(π₯)
22
π¦ = ππππ‘ππβ(π₯)
π ππ₯
DERIVADA COMPUESTA
1
ππ₯
ππππ (π₯)= π₯ ππππ (π) ππ₯
π 1 ππ₯ πΏπ(π₯) = ππ₯ π₯ ππ₯ π π₯ ππ₯ π = π π₯ πΏπ(π) ππ₯ ππ₯ π π₯ ππ₯ π = ππ₯ ππ₯ ππ₯ π ππ₯
ππ₯
π ππ(π₯ ) = cos(π₯) ππ₯
π ππ₯ πππ (π₯ ) = βπ ππ(π₯) ππ₯ ππ₯ π ππ₯
1
ππ₯
π‘ππ(π₯ ) = πππ 2π₯ ππ₯
π ππ₯ πππ‘(π₯ ) = βππ π 2 (π₯) ππ₯ ππ₯ π ππ₯ π ππ (π₯ ) = sec(π₯) tan(π₯) ππ₯ ππ₯ π ππ₯ ππ π (π₯ ) = βππ π(π₯)πππ‘(π₯) ππ₯ ππ₯ π 1 ππ₯ ππππ ππ (π₯) = ππ₯ β1 β π₯ 2 ππ₯ π β1 ππ₯ ππππππ (π₯ ) = ππ₯ β1 β π₯ 2 ππ₯ π ππ₯
1
ππ₯
ππππ‘ππ(π₯ ) = 1+π₯ 2 ππ₯
π ππ₯ π ππβ(π₯) = πππ β(π₯) ππ₯ ππ₯ π ππ₯ πππ β(π₯ ) = π ππβ(π₯ ) ππ₯ ππ₯ π 1 ππ₯ π‘ππβ(π₯ ) = 2 ππ₯ πππ β (π₯) ππ₯ π 1 ππ₯ ππππ ππβ(π₯ ) = ππ₯ βπ₯ 2 + 1 ππ₯ π 1 ππ₯ ππππππ β(π₯ ) = 2 ππ₯ βπ₯ β 1 ππ₯ π 1 ππ₯ ππππ‘ππβ(π₯ ) = ππ₯ 1 β π₯ 2 ππ₯
π π (π (π₯ ))π = ππ (π₯ )πβ1 π(π₯) ππ₯ ππ₯ π 1 π ππππ (π(π₯ ))= π(π₯) ππππ (π) ππ₯ π(π₯) ππ₯ π 1 π πΏπ(π (π₯ )) = π(π₯) ππ₯ π (π₯ ) ππ₯ π π(π₯) π π = π π(π₯) πΏπ(π) π (π₯ ) ππ₯ ππ₯ π π(π₯) π π = π π(π₯) π(π₯) ππ₯ ππ₯ π π π ππ(π (π₯ )) = cos(π(π₯ )) π (π₯ ) ππ₯ ππ₯ π π πππ (π (π₯)) = βπ ππ(π(π₯ )) π (π₯ ) ππ₯ ππ₯ π 1 π π‘ππ(π (π₯ )) = π (π₯ ) ππ₯ πππ 2 (π(π₯ )) ππ₯ π π πππ‘(π (π₯ )) = βππ π 2 (π (π₯)) π (π₯ ) ππ₯ ππ₯ π π π ππ(π (π₯ )) = sec(π (π₯ )) tan(π(π₯ )) π(π₯ ) ππ₯ ππ₯ π π ππ π(π (π₯)) = βππ π(π(π₯ ))πππ‘(π(π₯ )) π (π₯ ) ππ₯ ππ₯ π 1 π ππππ ππ(π (π₯ )) = π (π₯ ) ππ₯ β1 β (π(π₯ ))2 ππ₯ π β1 π ππππππ (π (π₯ )) = π (π₯ ) 2 ππ₯ ππ₯ β1 β (π (π₯)) π 1 π ππππ‘ππ(π (π₯)) = π (π₯ ) ππ₯ 1 + (π(π₯ ))2 ππ₯ π π π ππβ(π (π₯ )) = πππ β(π(π₯ )) π(π₯ ) ππ₯ ππ₯ π π πππ β(π(π₯ )) = π ππβ(π (π₯ )) π (π₯ ) ππ₯ ππ₯ π 1 π π‘ππβ(π (π₯ )) = π (π₯ ) 2 ππ₯ πππ β (π(π₯ )) ππ₯ π 1 π ππππ ππβ(π (π₯ )) = π (π₯ ) ππ₯ β(π (π₯ ))2 + 1 ππ₯ π 1 π ππππππ β(π (π₯ )) = π (π₯ ) ππ₯ β(π(π₯ ))2 β 1 ππ₯ π 1 π ππππ‘ππβ(π (π₯ )) = π (π₯ ) 2 ππ₯ 1 β (π(π₯)) ππ₯
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