Part 2/4:
The domain of the inverse tangent function is limited to ( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} ), ensuring that each input ( x ) corresponds to a unique ( y ) value. This not only simplifies our computations but also provides a clearer graphical representation. The graph of the tangent function experiences asymptotic behavior as it approaches ( \frac{\pi}{2} ) and ( -\frac{\pi}{2} ). Consequently, the graph of the inverse tangent function is the reflection of the function ( y = x ).
Implicit Differentiation of the Inverse Function
To find the derivative of ( y = \text{arctan}(x) ), we employ implicit differentiation. Starting from the equation
[
\tan(y) = x,
]
we differentiate both sides with respect to ( x ). Utilizing the chain rule, we find that:
[