Tabla de Derivadas e Integrales

Función

Derivada

Integral

y = c

y’ = 0

c.x

y = c.x

y’ = c

c.x²/2

y = xⁿ

y’ = n.x^n-1

xⁿ⁺¹/n+1

y = x^-n

y’ = -1/(n.x^n-1)

x-ⁿ⁺¹/-n+1

y = x^½

y’ = 1/(2.x^½)

2.x^3/2/3

y = x^a/b

y’ = a.x^(a/b)-1/b

x^(a/b)+1/[(a/b)+1]

y = 1/x

y’ = -1/x²

ln x

y = sen x

y’ = cos x

-cos x

y = cos x

y’ = -sen x

sen x

y = tg x

y’ = 1/cos²x

-ln cos x

y = cotg x

y’ = -1/sen²x

ln sen x

y = sec x

y’ = sen x/cos²x

ln (tg ½.x)

y = cosec x

y’ = -cos x/sen²x

ln [cos x/(1 - sen x)]

y = arcsen x

y’ = 1/(1 – x²)^½

x.arcsen x + (1 – x²)^½

y = arccos x

y’ = -1/(1 – x²)^½

x.arccos x – (1 – x²)^½

y = arctg x

y’ = 1/(1 + x²)

x.arctg x – ½ln (1 + x²)

y = arccotg x

y’ = -1/(1 + x²)

x.arccotg x + ½ln (1 + x²)

y = arcsec x

y’ = 1/[x.(x² -1)^½]

1

y = arccosec x

y’ = -1/[x.(x² – 1)^½]

2

y = senh x

y’ = cosh x

cosh x

y = cosh x

y’ = senh x

senh x

y = tgh x

y’ = sech²x

ln cosh x

y = cotgh x

y’ = -cosech²x

ln senh x

y = sech x

y’ = -sech x.tgh x

3

y = cosech x

y’ = -cosech x.cotgh x

4

y = ln x

y’ = 1/x

x.(ln x – 1)

y = log_ax

y’ = 1/x.ln a

x.( log_ax – 1/ln a)

y = e^x

y’ = e^x

e^x

y = a^x

y’ = a^x.ln a

a^x/ln a

y = x^x

y’ = x^x.(ln x + 1)

5

y = e^u

y’ = e^u.u’

6

y = u.v

y’ = u’.v + v’.u

òu.dv + òv.du

y = u/v

y’ = (u’.v – v’.u)/v²

7

y = u^v

y’ = u^v.(v’.lnu + v.u’/u)

8

y = ln_uv

y’ = (v’.u.lnu – u’.v.lnv)/v.u.ln²u

9