Question

Find the second order derivatives of the function
tan⁻¹x

Answer 1

Answer:

Let:

y

=

arctan

x

so that:

x

=

tan

y

differentiate this last equality with respect to

x

:

1

=

sec

2

y

d

y

d

x

Now using the trigonometric inequality:

sec

2

y

=

1

+

tan

2

y

we have:

1

=

(

1

+

tan

2

y

)

d

y

d

x

1

=

(

1

+

x

2

)

d

y

d

x

that is:

d

y

d

x

=

1

1

+

x

2

Differentiate again using the chain rule:

d

2

y

d

x

2

=

d

d

x

1

1

+

x

2

d

2

y

d

x

2

=

−

1

(

1

+

x

2

)

2

d

d

x

(

1

+

x

2

)

d

2

y

d

x

2

=

−

2

x

(

1

+

x

2

)

2