Higher Engineering Mathematics, Sixth Edition

664 Higher Engineering Mathematics

yorf(x)

dy

dx

orf′(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

yorf(x)

dy

dx

orf′(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:

Wheny=uvanduandvare functions ofxthen:

dy

dx

=u

dv

dx

+v

du

dx

Quotient rule:

Wheny=

u

v

anduandvare functions ofxthen:

dy

dx

=

v

du

dx

−u

dv

dx

v^2

Function of a function:

Ifuis a function ofxthen:

dy

dx

=

dy

du

×

du

dx

Parametric differentiation:

Ifxandyare both functions ofθ, 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