Aula 6 - Derivadas De Ordem Superior
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Derivadas de Ordem Superior
y = f(x) f ′ (x) =
dy = y′ dx
f ′′ (x) =
d dy d 2 y '' = 2 =y dx dx dx
f ′′′ (x) =
d d2 y d3 y 2 = 3 = y ''' dx dx dx
derivada primeira
derivada segunda
derivada terceira
De um modo geral
f n (x) =
dn y = yn n dx
Exemplos:
Calcular y ′ , y ′′ e y ′′′ :
a) y = x 8 − 4 x 4 + 2 x
y ' = 8 x 7 − 16 x 3 + 2 y " = 56 x 6 − 48 x 2 y "' = 336 x 5 − 96 x
b) y = 4 x 2 − 2 x + 40 x 3 − x
1
1 − y = 8 x − 2 + 120 x − x 2 2 '
2
3
1 − y = 8 + 240 x + x 2 4 ''
5
y ''' = 240 −
3 −2 x 8
y = f(x) f ′ (x) =
dy = y′ dx
f ′′ (x) =
d dy d 2 y '' = 2 =y dx dx dx
f ′′′ (x) =
d d2 y d3 y 2 = 3 = y ''' dx dx dx
derivada primeira
derivada segunda
derivada terceira
De um modo geral
f n (x) =
dn y = yn n dx
Exemplos:
Calcular y ′ , y ′′ e y ′′′ :
a) y = x 8 − 4 x 4 + 2 x
y ' = 8 x 7 − 16 x 3 + 2 y " = 56 x 6 − 48 x 2 y "' = 336 x 5 − 96 x
b) y = 4 x 2 − 2 x + 40 x 3 − x
1
1 − y = 8 x − 2 + 120 x − x 2 2 '
2
3
1 − y = 8 + 240 x + x 2 4 ''
5
y ''' = 240 −
3 −2 x 8
