664 Higher Engineering Mathematics
y or f (x)
dy
dx
or f (x)
cos −1 f (x)
− f (x)
1 − [ f (x)] 2
tan −1 x
a
a
a 2 + x 2
tan −1 f (x)
f (x)
1 + [ f (x)] 2
sec −1 x
a
a
x
√
x 2 − a 2
sec −1 f (x)
f (x)
f (x)
[ f (x)] 2 − 1
cosec −1 x
a
−a
x
√
x 2 − a 2
cosec −1 f (x)
− f (x)
f (x)
[ f (x)] 2 − 1
cot −1 x
a
−a
a 2 + x 2
cot −1 f (x)
− f (x)
1 + [ f (x)] 2
sinh −1 x
a
1
√
x 2 + a 2
sinh −1 f (x)
f (x)
[ f (x)] 2 + 1
cosh −1 x
a
1
√
x 2 − a 2
cosh −1 f (x)
f (x)
[ f (x)] 2 − 1
tanh −1 x
a
a
a 2 − x 2
tanh −1 f (x)
f (x)
1 − [ f (x)] 2
sech −1 x
a
−a
x
√
a 2 − x 2
sech −1 f (x)
− f (x)
f (x)
1 − [ f (x)] 2
y or f (x)
dy
dx
or f (x)
cosech
−1 x
a
−a
x
√
x 2 + a 2
cosech −1 f (x)
− f (x)
f (x)
[ f (x)] 2 + 1
coth −1 x
a
a
a 2 − x 2
coth
−1 f (x)
f (x)
1 − [ f (x)] 2
Product rule:
When y = uv and u and v are functions of x then:
dy
dx
= u
dv
dx
+ v
du
dx
Quotient rule:
When y =
u
v
and u and v are functions of x then:
dy
dx
=
v
du
dx
− u
dv
dx
v 2
Function of a function:
If u is a function of x then:
dy
dx
=
dy
du
×
du
dx
Parametric differentiation:
If x and y are both functions of θ, then:
dy
dx
=
dy
dθ
dx
dθ
and
d 2 y
dx 2 =
d
dθ
dy
dx
dx
dθ
Implicit function:
d
dx
[ f (y)]=
d
dy
[ f (y)] ×
dy
dx