Formulario Cálculo Diferencial Matefísica AR PDF

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Formulario Cálculo DiferencialDefinición de la derivada o Regla de los 4 pasoslimℎ→𝑓(𝑥 +ℎ)−𝑓(𝑥)ℎ; limΔ𝑥→𝑓(𝑥 +Δ𝑥)−𝑓(𝑥)Δ𝑥Derivadas de funciones algebraicas1.𝑑𝑑𝑥[𝑐]= 02.𝑑𝑑𝑥[𝑥]= 13.𝑑𝑑𝑥[𝑐𝑥]= 𝑐4.𝑑𝑑𝑥[𝑢 ±𝑣 ±𝑤]=𝑑𝑑𝑥[𝑢]±𝑑𝑑𝑥[𝑣]±𝑑𝑑𝑥[𝑤]5.𝑑𝑑𝑥[𝑥𝑛]= 𝑛𝑥𝑛−6.𝑑𝑑𝑥[𝑥𝑚𝑛 ] =𝑚𝑛𝑥𝑚−𝑛𝑛7.𝑑𝑑𝑥[𝑐𝑥𝑛]= 𝑐 𝑛𝑥𝑛−8.𝑑𝑑𝑥[1𝑥𝑛] = −𝑛𝑥𝑛+9.𝑑𝑑𝑥[𝑢...

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Formula Formulario rio Cálculo Diferenci Diferencial al Definición de la derivada o Regla de los 4 pasos

𝑓(𝑥 + ℎ) − 𝑓(𝑥) 𝑓(𝑥 + Δ𝑥) − 𝑓(𝑥) ; lim ℎ→0 Δ𝑥→0 Δ𝑥 ℎ lim

Derivadas de funciones algebraicas

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

[𝑐 ] = 0 [𝑥 ] = 1

𝑑𝑥

12.

[𝑐𝑥 ] = 𝑐 [𝑢 ± 𝑣 ± 𝑤] = [𝑥 𝑛 ] = 𝑛𝑥 𝑛−1 𝑚

[𝑥 𝑛 ] = [𝑐𝑥 1

𝑛]

𝑚 𝑛

𝑥

𝑑

𝑑𝑥

𝑑

13.

𝑑

[𝑢] ± [𝑣 ] ± [𝑤] 𝑑𝑥 𝑑𝑥

14. 15.

𝑚−𝑛 𝑛

= 𝑐𝑛𝑥

[ ]=−

𝑑𝑥 𝑥 𝑛 𝑑 𝑑𝑥 𝑑

11.

16.

𝑛−1

17.

𝑛

𝑥 𝑛+1 𝑑𝑣

[𝑢 ∙ 𝑣] = 𝑢

𝑑𝑥

18.

𝑑𝑢 𝑑𝑥 𝑑𝑤

+𝑣

[𝑢 ∙ 𝑣 ∙ 𝑤] = 𝑢𝑣

𝑑𝑥

𝑑𝑣

+ 𝑢𝑤

𝑑𝑥

+

19.

𝑑𝑢 𝑣𝑤 𝑑𝑥

20.

𝑑

𝑢

[ ]=

𝑑𝑥 𝑣 𝑑 𝑢

[ ]=

𝑑𝑥 𝑐 𝑑 𝑐

𝑑𝑣

𝑑𝑢

𝑣 𝑑𝑥 − 𝑢 𝑑𝑥 𝑣2 𝑑𝑢 ∙ 𝑐 𝑑𝑥 𝑐 𝑑𝑣

1

[ ]=−

∙

𝑣 2 𝑑𝑥 2𝑎𝑏 [ ]= ( 𝑑𝑥 𝑎𝑥+𝑏 𝑎𝑥+𝑏)2 𝑎𝑥+𝑏 2𝑎𝑏 𝑑 [ ]=−( 𝑑𝑥 𝑎𝑥−𝑏 𝑎𝑥−𝑏)2 𝑑𝑦 𝑑𝑦 𝑑𝑢 = ∙ 𝑑𝑥 𝑑𝑥 𝑑𝑢 𝑑 𝑑𝑢 𝑛 [𝑢 ] = 𝑛𝑢 𝑛−1 ∙ 𝑑𝑥 𝑑𝑥 1 𝑑 𝑛 𝑑𝑢 𝑑𝑥 𝑣 𝑑 𝑎𝑥−𝑏

∙

[ √𝑢] =

𝑛 𝑑𝑥 𝑑𝑥 𝑛∙ √𝑢𝑛−1 𝑚 𝑛 𝑚−𝑛 𝑑 𝑛 𝑚 [ √𝑢 ] = 𝑛 ∙ √𝑢 𝑑𝑥 𝑢 𝑑𝑢 𝑑

𝑑𝑥

[|𝑢|] =

|𝑢|

∙

𝑑𝑥

Derivadas de funciones trigonométricas

1. 2. 3.

𝑑

𝑑𝑥 𝑑

[𝑠𝑒𝑛 𝑢 ] = cos 𝑢 ∙

𝑑𝑢

4.

𝑑𝑥 𝑑𝑢

[cos 𝑢] = −𝑠𝑒𝑛 𝑢 ∙ 𝑑𝑥 𝑑

𝑑𝑥

[tan 𝑢] = sec 2 𝑢 ∙

𝑑𝑥 𝑑𝑢 𝑑𝑥

5. 6.

𝑑

𝑑𝑥 𝑑

[cot 𝑢 ] = − csc 2 𝑢 ∙

𝑑𝑢

𝑑𝑥

𝑑𝑢

[sec 𝑢] = sec 𝑢 ∙ tan 𝑢 ∙ 𝑑𝑥 𝑑

𝑑𝑥

[csc 𝑢] = − csc 𝑢 cot 𝑢 ∙

𝑑𝑥 𝑑𝑢

𝑑𝑥

Derivadas de funciones trigonométricas inversas

1. 2. 3.

𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥

[𝑎𝑟𝑐𝑠𝑒𝑛 𝑢] = [arccos 𝑢] = [arctan 𝑢] =

1

∙ 2

𝑑𝑢

𝑑𝑥 𝑑𝑢 − ∙ 𝑑𝑥 2 √1−𝑢 1 𝑑𝑢 √1−𝑢 1

1+𝑢2

∙

𝑑𝑥

4. 5. 6.

𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥

1

𝑑𝑢

1

∙

[arccot 𝑢] = − ∙ 1+𝑢2

[arcsec 𝑢] =

|𝑢|√𝑢2 −1 1

[arccsc 𝑢] = −

𝑑𝑥 𝑑𝑢

|𝑢|√𝑢2 −1

𝑑𝑥 𝑑𝑢

∙

𝑑𝑥

𝑑𝑢

∙ 𝑑𝑥

Derivadas de funciones exponenciales 𝑑

𝑢]

[𝑒 = 𝑒 ∙ 𝑑𝑥 𝑑 2. 𝑑𝑥 [𝑒 𝑥 ] = 𝑒 𝑥

1.

3. 4.

𝑢

𝑑 [𝑥 𝑥 ] = 𝑑𝑥 𝑑 𝑠𝑒𝑛 𝑥 𝑑𝑥

[𝑥

𝑑𝑢 𝑑𝑥

5. 6.

𝑥 𝑥 (1 + ln 𝑥 )

]=𝑥

𝑠𝑒𝑛 𝑥

7.

(cos 𝑥 ∙ ln 𝑥 +

𝑠𝑒𝑛 𝑥 𝑥

)

𝑑

[𝑎𝑢 ] = 𝑎𝑢 ∙ ln(𝑎) ∙ 𝑑𝑥 𝑑 [𝑎 𝑥 ] = 𝑎 𝑥 ∙ ln (𝑎)

𝑑𝑥 𝑑

𝑑𝑥

[𝑢𝑣 ] = 𝑣𝑢 𝑣−1 ∙

𝑑𝑢

𝑑𝑥

𝑑𝑢 𝑑𝑥

+ 𝑢𝑣 ∙ ln 𝑢 ∙

𝑑𝑣

𝑑𝑥

Derivadas de funciones logarítmicas

1. 2.

𝑑

𝑑𝑥 𝑑

𝑑𝑥

[log 𝑎 𝑢] = [log 𝑎 𝑥 ] =

1 𝑑𝑢 ∙ ln(𝑎) ∙ 𝑢 𝑑𝑥 1

→ 𝑎 ≠ 0,1

3. 4.

ln(𝑎) ∙ 𝑥

𝑑

𝑑𝑥 𝑑

𝑑𝑥

[ln 𝑢] = [ln 𝑥 ] =

1

𝑢 1

∙

𝑑𝑢

𝑑𝑥

𝑥

Derivadas de funciones hiperbólicas

1. 2. 3.

𝑑

𝑑𝑢 𝑑𝑥 𝑑 𝑑𝑢 [cosh 𝑢] = 𝑠𝑒𝑛ℎ 𝑢 ∙ 𝑑𝑥 𝑑𝑥 𝑑 𝑑𝑢 [tanh 𝑢 ] = sech2 𝑢 ∙ 𝑑𝑥 𝑑𝑥

[𝑠𝑒𝑛ℎ 𝑢] = cosh 𝑢 ∙ 𝑑𝑥

4. 5. 6.

𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥

𝑑𝑢 𝑑𝑥

[coth 𝑢 ] = − csch2 𝑢 ∙

[sech 𝑢] = − sech 𝑢 ∙ tanh 𝑢 ∙ [csch 𝑢] = − csch 𝑢 ∙ coth 𝑢 ∙

Derivadas de funciones hiperbólicas inversas

1. 2. 3.

𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥

[𝑎𝑟𝑐𝑠𝑒𝑛ℎ 𝑢 ] = [arccosh 𝑢] = [arctanh 𝑢] =

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1

∙

𝑑𝑢

√𝑢2 +1 𝑑𝑥 ±1 𝑑𝑢

∙

√𝑢2 −1 𝑑𝑥 1 𝑑𝑢 ∙ 1−𝑢2 𝑑𝑥

4. 5. 6.

𝑑

𝑑𝑥 𝑑

𝑑𝑥 𝑑

𝑑𝑥

[arccoth 𝑢] = [arcsech 𝑢] =

1

1−𝑢2

∙

∓1

𝑑𝑢

𝑑𝑥

|𝑢|√1−𝑢2

[arccsch 𝑢] = −

1

∙

𝑑𝑢

|𝑢|√1+𝑢2

𝑑𝑥 𝑑𝑢

∙

𝑑𝑥

𝑑𝑢 𝑑𝑥 𝑑𝑢 𝑑𝑥...