Question

Differentiate tan^-1(√1-x/1+x)​

Answer 1

Answer:

−

1

2

√

1

−

x

2

,

−

1

<

x

<

1

.

Explanation:

Let,

y

=

a

r

c

tan

√

1

−

x

1

+

x

.

Now, note that,

√

1

−

x

1

+

x

will be meaningful, iff,

(

1

−

x

)

≥

0

,

and

(

1

+

x

)

>

0

,

i

.

e

.

,

−

1

<

x

≤

1

.

Also, the Range of

cos

function is

[

−

1

,

1

]

.

We can, hence, subst.,

x

=

cos

θ

,

so that,

θ

=

a

r

c

cos

x

.

Then, we have,

y

=

a

r

c

tan

√

1

−

cos

θ

1

+

cos

θ

,

=

a

r

c

tan











⎷

2

sin

2

(

θ

2

)

2

cos

2

(

θ

2

)

,

=

a

r

c

tan

(

tan

(

θ

2

)

)

,

=

θ

2

.

∴

y

=

1

2

a

r

c

cos

x

,

∴

d

y

d

x

=

1

2

⋅

−

1

√

1

−

x

2

=

−

1

2

√

1

−

x

2

,

−

1

<

x

<

1