Question

Find y' if x = tan(y)

Answer 1

Use the chain rule:


`x=\tan y`

`(\mathrm d)/(\mathrm dx)x=(\mathrm d)/(\mathrm dx)\tan y`

`1=\sec^2y(\mathrm dy)/(\mathrm dx)`

`1=\sec^2y\,y'`

`y'=\frac1{\sec^2y}=\cos^2y`

Furthermore, if you restrict the domain of
`y`, we could write


`x=\tan y\iff \tan^(-1)x=y`

and so we could also have


`y'=\cos^2(\tan^(-1)x)=\left(\frac1{√(x^2+1)}\right)^2=\frac1{x^2+1}`