Tablas De Ayuda De Derivadas.pdf

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ÁLGEBRA DE DERIVADAS

𝑑 𝑑 𝑑 (𝑓(π‘₯) Β± 𝑔(π‘₯)) = 𝑓(π‘₯) Β± 𝑔(π‘₯) 𝑑π‘₯ 𝑑π‘₯ 𝑑π‘₯

𝑑 π‘˜π‘“(π‘₯) 𝑑π‘₯

𝑑 𝑑 𝑑 (𝑓(π‘₯)𝑔(π‘₯)) = 𝑓(π‘₯) ( 𝑔(π‘₯)) + ( 𝑓(π‘₯)) 𝑔(π‘₯) 𝑑π‘₯ 𝑑π‘₯ 𝑑π‘₯

𝑑 𝑓(π‘₯) ( )= 𝑑π‘₯ 𝑔(π‘₯)

=π‘˜

𝑑 𝑓(π‘₯); 𝑑π‘₯

𝑔(π‘₯) (

π‘˜: πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘π‘’

𝑑 𝑑 𝑓(π‘₯)) βˆ’ 𝑓(π‘₯) ( 𝑔(π‘₯)) 𝑑π‘₯ 𝑑π‘₯ [𝑔(π‘₯)]2

𝑑 𝑔(π‘₯) 𝑑 𝑑 (𝑓(π‘₯)𝑔(π‘₯) ) = 𝑓(π‘₯)𝑔(π‘₯) [ ( 𝑓(π‘₯)) + 𝐿𝑛(𝑓(π‘₯)) ( 𝑔(π‘₯))] 𝑑π‘₯ 𝑓(π‘₯) 𝑑π‘₯ 𝑑π‘₯ TABLA DE DERIVADAS

NΒΊ

FUNCIΓ“N

DERIVADA

1

𝑦 = π‘˜; π‘˜: πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘π‘’

2

𝑦=π‘₯

3

𝑦 = π‘₯𝑛

π‘‘π‘˜ =0 𝑑π‘₯ 𝑑π‘₯ =1 𝑑π‘₯ 𝑑 𝑛 𝑑π‘₯ π‘₯ = 𝑛π‘₯ π‘›βˆ’1 𝑑π‘₯ 𝑑π‘₯

4

𝑦 = π‘™π‘œπ‘”π‘Ž (π‘₯)

5

𝑦 = 𝐿𝑛(π‘₯)

6

𝑦 = π‘Žπ‘₯

7

𝑦 = 𝑒π‘₯

8

𝑦 = 𝑠𝑒𝑛(π‘₯)

9

𝑦 = π‘π‘œπ‘ (π‘₯)

10

𝑦 = π‘‘π‘Žπ‘›(π‘₯)

11

𝑦 = π‘π‘œπ‘‘(π‘₯)

12

𝑦 = 𝑠𝑒𝑐(π‘₯)

13

𝑦 = 𝑐𝑠𝑐(π‘₯)

14

𝑦 = π‘Žπ‘Ÿπ‘π‘ π‘’π‘›(π‘₯)

15

𝑦 = π‘Žπ‘Ÿπ‘π‘π‘œπ‘ (π‘₯)

16

𝑦 = π‘Žπ‘Ÿπ‘π‘‘π‘Žπ‘›(π‘₯)

17

𝑦 = π‘ π‘’π‘›β„Ž(π‘₯)

18

𝑦 = π‘π‘œπ‘ β„Ž(π‘₯)

19

𝑦 = π‘‘π‘Žπ‘›β„Ž(π‘₯)

20

𝑦 = π‘Žπ‘Ÿπ‘π‘ π‘’π‘›β„Ž(π‘₯)

21

𝑦 = π‘Žπ‘Ÿπ‘π‘π‘œπ‘ β„Ž(π‘₯)

22

𝑦 = π‘Žπ‘Ÿπ‘π‘‘π‘Žπ‘›β„Ž(π‘₯)

𝑑 𝑑π‘₯

DERIVADA COMPUESTA

1

𝑑π‘₯

π‘™π‘œπ‘”π‘Ž (π‘₯)= π‘₯ π‘™π‘œπ‘”π‘Ž (𝑒) 𝑑π‘₯

𝑑 1 𝑑π‘₯ 𝐿𝑛(π‘₯) = 𝑑π‘₯ π‘₯ 𝑑π‘₯ 𝑑 π‘₯ 𝑑π‘₯ π‘Ž = π‘Ž π‘₯ 𝐿𝑛(π‘Ž) 𝑑π‘₯ 𝑑π‘₯ 𝑑 π‘₯ 𝑑π‘₯ 𝑒 = 𝑒π‘₯ 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑π‘₯

𝑑π‘₯

𝑠𝑒𝑛(π‘₯ ) = cos(π‘₯) 𝑑π‘₯

𝑑 𝑑π‘₯ π‘π‘œπ‘ (π‘₯ ) = βˆ’π‘ π‘’π‘›(π‘₯) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑π‘₯

1

𝑑π‘₯

π‘‘π‘Žπ‘›(π‘₯ ) = π‘π‘œπ‘  2π‘₯ 𝑑π‘₯

𝑑 𝑑π‘₯ π‘π‘œπ‘‘(π‘₯ ) = βˆ’π‘π‘ π‘ 2 (π‘₯) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑π‘₯ 𝑠𝑒𝑐 (π‘₯ ) = sec(π‘₯) tan(π‘₯) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑π‘₯ 𝑐𝑠𝑐 (π‘₯ ) = βˆ’π‘π‘ π‘(π‘₯)π‘π‘œπ‘‘(π‘₯) 𝑑π‘₯ 𝑑π‘₯ 𝑑 1 𝑑π‘₯ π‘Žπ‘Ÿπ‘π‘ π‘’π‘› (π‘₯) = 𝑑π‘₯ √1 βˆ’ π‘₯ 2 𝑑π‘₯ 𝑑 βˆ’1 𝑑π‘₯ π‘Žπ‘Ÿπ‘π‘π‘œπ‘ (π‘₯ ) = 𝑑π‘₯ √1 βˆ’ π‘₯ 2 𝑑π‘₯ 𝑑 𝑑π‘₯

1

𝑑π‘₯

π‘Žπ‘Ÿπ‘π‘‘π‘Žπ‘›(π‘₯ ) = 1+π‘₯ 2 𝑑π‘₯

𝑑 𝑑π‘₯ π‘ π‘’π‘›β„Ž(π‘₯) = π‘π‘œπ‘ β„Ž(π‘₯) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑π‘₯ π‘π‘œπ‘ β„Ž(π‘₯ ) = π‘ π‘’π‘›β„Ž(π‘₯ ) 𝑑π‘₯ 𝑑π‘₯ 𝑑 1 𝑑π‘₯ π‘‘π‘Žπ‘›β„Ž(π‘₯ ) = 2 𝑑π‘₯ π‘π‘œπ‘ β„Ž (π‘₯) 𝑑π‘₯ 𝑑 1 𝑑π‘₯ π‘Žπ‘Ÿπ‘π‘ π‘’π‘›β„Ž(π‘₯ ) = 𝑑π‘₯ √π‘₯ 2 + 1 𝑑π‘₯ 𝑑 1 𝑑π‘₯ π‘Žπ‘Ÿπ‘π‘π‘œπ‘ β„Ž(π‘₯ ) = 2 𝑑π‘₯ √π‘₯ βˆ’ 1 𝑑π‘₯ 𝑑 1 𝑑π‘₯ π‘Žπ‘Ÿπ‘π‘‘π‘Žπ‘›β„Ž(π‘₯ ) = 𝑑π‘₯ 1 βˆ’ π‘₯ 2 𝑑π‘₯

𝑑 𝑑 (𝑓 (π‘₯ ))𝑛 = 𝑛𝑓 (π‘₯ )π‘›βˆ’1 𝑓(π‘₯) 𝑑π‘₯ 𝑑π‘₯ 𝑑 1 𝑑 π‘™π‘œπ‘”π‘Ž (𝑓(π‘₯ ))= 𝑓(π‘₯) π‘™π‘œπ‘”π‘Ž (𝑒) 𝑑π‘₯ 𝑓(π‘₯) 𝑑π‘₯ 𝑑 1 𝑑 𝐿𝑛(𝑓 (π‘₯ )) = 𝑓(π‘₯) 𝑑π‘₯ 𝑓 (π‘₯ ) 𝑑π‘₯ 𝑑 𝑓(π‘₯) 𝑑 π‘Ž = π‘Ž 𝑓(π‘₯) 𝐿𝑛(π‘Ž) 𝑓 (π‘₯ ) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑓(π‘₯) 𝑑 𝑒 = 𝑒 𝑓(π‘₯) 𝑓(π‘₯) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑 𝑠𝑒𝑛(𝑓 (π‘₯ )) = cos(𝑓(π‘₯ )) 𝑓 (π‘₯ ) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑 π‘π‘œπ‘ (𝑓 (π‘₯)) = βˆ’π‘ π‘’π‘›(𝑓(π‘₯ )) 𝑓 (π‘₯ ) 𝑑π‘₯ 𝑑π‘₯ 𝑑 1 𝑑 π‘‘π‘Žπ‘›(𝑓 (π‘₯ )) = 𝑓 (π‘₯ ) 𝑑π‘₯ π‘π‘œπ‘  2 (𝑓(π‘₯ )) 𝑑π‘₯ 𝑑 𝑑 π‘π‘œπ‘‘(𝑓 (π‘₯ )) = βˆ’π‘π‘ π‘ 2 (𝑓 (π‘₯)) 𝑓 (π‘₯ ) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑 𝑠𝑒𝑐(𝑓 (π‘₯ )) = sec(𝑓 (π‘₯ )) tan(𝑓(π‘₯ )) 𝑓(π‘₯ ) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑 𝑐𝑠𝑐(𝑓 (π‘₯)) = βˆ’π‘π‘ π‘(𝑓(π‘₯ ))π‘π‘œπ‘‘(𝑓(π‘₯ )) 𝑓 (π‘₯ ) 𝑑π‘₯ 𝑑π‘₯ 𝑑 1 𝑑 π‘Žπ‘Ÿπ‘π‘ π‘’π‘›(𝑓 (π‘₯ )) = 𝑓 (π‘₯ ) 𝑑π‘₯ √1 βˆ’ (𝑓(π‘₯ ))2 𝑑π‘₯ 𝑑 βˆ’1 𝑑 π‘Žπ‘Ÿπ‘π‘π‘œπ‘ (𝑓 (π‘₯ )) = 𝑓 (π‘₯ ) 2 𝑑π‘₯ 𝑑π‘₯ √1 βˆ’ (𝑓 (π‘₯)) 𝑑 1 𝑑 π‘Žπ‘Ÿπ‘π‘‘π‘Žπ‘›(𝑓 (π‘₯)) = 𝑓 (π‘₯ ) 𝑑π‘₯ 1 + (𝑓(π‘₯ ))2 𝑑π‘₯ 𝑑 𝑑 π‘ π‘’π‘›β„Ž(𝑓 (π‘₯ )) = π‘π‘œπ‘ β„Ž(𝑓(π‘₯ )) 𝑓(π‘₯ ) 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑 π‘π‘œπ‘ β„Ž(𝑓(π‘₯ )) = π‘ π‘’π‘›β„Ž(𝑓 (π‘₯ )) 𝑓 (π‘₯ ) 𝑑π‘₯ 𝑑π‘₯ 𝑑 1 𝑑 π‘‘π‘Žπ‘›β„Ž(𝑓 (π‘₯ )) = 𝑓 (π‘₯ ) 2 𝑑π‘₯ π‘π‘œπ‘ β„Ž (𝑓(π‘₯ )) 𝑑π‘₯ 𝑑 1 𝑑 π‘Žπ‘Ÿπ‘π‘ π‘’π‘›β„Ž(𝑓 (π‘₯ )) = 𝑓 (π‘₯ ) 𝑑π‘₯ √(𝑓 (π‘₯ ))2 + 1 𝑑π‘₯ 𝑑 1 𝑑 π‘Žπ‘Ÿπ‘π‘π‘œπ‘ β„Ž(𝑓 (π‘₯ )) = 𝑓 (π‘₯ ) 𝑑π‘₯ √(𝑓(π‘₯ ))2 βˆ’ 1 𝑑π‘₯ 𝑑 1 𝑑 π‘Žπ‘Ÿπ‘π‘‘π‘Žπ‘›β„Ž(𝑓 (π‘₯ )) = 𝑓 (π‘₯ ) 2 𝑑π‘₯ 1 βˆ’ (𝑓(π‘₯)) 𝑑π‘₯

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