Question

differentiate y=x^tan^-1x​

Answer 1

z

=

tan

−

1

(

y

x

)

. The answers are

∂

z

∂

x

=

−

y

x

2

+

y

2

and

∂

z

∂

y

=

x

x

2

+

y

2

.

Both of these facts can be derived with the Chain Rule, the Power Rule, and the fact that

y

x

=

y

x

−

1

as follows:

∂

z

∂

x

=

1

1

+

(

y

x

)

2

⋅

∂

∂

x

(

y

x

−

1

)

=

1

1

+

(

y

x

)

2

⋅

(

−

y

x

−

2

)

=

−

y

x

2

+

y

2

and

∂

z

∂

y

=

1

1

+

(

y

x

)

2

⋅

∂

∂

y

(

y

x

−

1

)

=

1

1

+

(

y

x

)

2

⋅

(

x

−

1

)

=

1

x

1

+

y

2

x

2

=

x

x

2

+

y

2