Differentiate with respect to x

[ZedCar]

Homework Statement

Differentiate with respect to x

y = tan^(-1) x

Homework Equations

The Attempt at a Solution

ANSWER in book:

y = tan^(-1) x
x = tan y
dx/dy = sec^(2) y = (1 + x^2)
dy/dx = 1 / (1 + x^2)

How is it known that sec^(2) y = (1 + x^2)
Does it follow from an identity?

 

[ZedCar]

Have this sorted.

It follows from the identity sin^2 (x) + cos^2 (x) = 1

 

[SammyS]

Homework Statement

Differentiate with respect to x

y = tan^(-1) x

Homework Equations

The Attempt at a Solution

ANSWER in book:

y = tan^(-1) x
x = tan y
dx/dy = sec^(2) y = (1 + x^2)
dy/dx = 1 / (1 + x^2)

How is it known that sec^(2) y = (1 + x^2)
Does it follow from an identity?

It follows from

x = tan(y)​

So that

tan²(y)=x²​

Then use the identity relating sec² and tan² , etc.