What is an exampleof computing the derivative?
The following solves the derivative at xafor the function f(x) x^2 , using the defin-
ition for a derivative (see above):
Are there higher derivativesof a function?
Yes, there are higher derivatives of a function, often referred to as higher order deriva-
tives. The “initial” derivative is often written as f' (x), but the “ ' ” is assumed in most
cases. The next derivative is the second derivative (second order derivative), most
often written as f' (x); the next is the third derivative, or third order derivative, most
often written as f''' (x); fourth derivative, or fourth order derivative, most often writ-
ten as f(4)(x); and so on. The notation for the higher derivatives, or the nth derivative,
is as follows:
d/dx(dn^1 y/dxn^1 )
dny/dxn
f(n)(x)
y(n)
0
lim ()()
h
fa h fa
2 a
+-
h" =
()()()
h
fa h fa
h
ah a
h
ah h
ah
2
2
+-^222
=
+-
=
+
=+
dx
d
v
u
v
vudxd dxdvu
= 2
dx
d u
dx
dv v
dx
]gxu =+du
dx
d cc
dx
]gu = du
dx
d v
dx
d
dx
]gu!!= u dv
dx
d
] ng=nxx n-^1
dx
d
]xg 1=
dx
d]cg 0=
224
