Question

if x=1/y then show that dx/dy=-√(1+x^4/1+y^4)​

Answer 1

Step-by-step explanation:

y=log[

(1+x)

−

(1−x)

(1+x)

+

(1−x)

]

1/2

Put x=sin2θ

y=log[

(1+sin2θ)

−

(1−sin2θ)

(1+sin2θ)

+

(1−sin2θ)

]

1/2

∵

1±sin2θ

=cosθ±sinθ

y=log[

(cosθ+sinθ)−(cosθ−sinθ)

(cosθ+sinθ)+(cosθ−sinθ)

]

1/2

∴y=

2

1

log(

2sinθ

2cosθ

)

y=

2

1

logcotθ

∴

dθ

dy

=

2

1

cotθ

1

.(−cosec

2

θ)=−

2cosθsinθ

1

=−

sin2θ

1

∴

dx

dy

=

dθ

dy

.

dx

dθ

=−

sin2θ

1

.

2cos2θ

1

(∵x=sin2θ⇒

dθ

dx

=2cos2θ)

=−

2x

1−x

2

1