91Ó°ÊÓ

The chain rule is an essential mathematical tool for finding derivatives of composite functions. When dealing with functions nested within another, like in our example with \((\tan^{-1}x)^2\), the chain rule lets us break down the problem. The idea is simple:

• Differentiate the outer function, treating the inner function as a single variable.
• Then, multiply this by the derivative of the inner function.

This two-step approach helps manage the complexity of the differentiation process. Let's see how this works in the exercise:
Given the function \( y = (\tan^{-1}x)^2 \), we first set \( u = \tan^{-1}x \), which makes the outer function \( y = u^2 \). This allows us to use the chain rule by finding \( \frac{dy}{du} = 2u \). Then, we also need to find \( \frac{du}{dx} = \frac{1}{1+x^2} \). Finally, use the chain rule formula \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \) and substitute back for \( u \), resulting in
\[ \frac{dy}{dx} = 2\tan^{-1}x \cdot \frac{1}{1+x^2}. \]

Inverse trigonometric functions are critical in many mathematical applications, especially in calculus. These functions provide solutions for angles when given certain trigonometric values. In this exercise, the inverse tangent function, or \(\tan^{-1}x\), is involved, which reverses the tangent by returning the angle whose tangent is \(x\).
The differentiation of inverse trig functions can be slightly trickier than their regular counterparts. For instance, the derivative of \(\tan^{-1}x\) is \(\frac{1}{1+x^2}\). This formula is fundamental when applying the chain rule since it serves as the derivative of the inner function in our example. Knowing these derivatives is crucial:

• \( \frac{d}{dx} \tan^{-1}x = \frac{1}{1+x^2} \)
• \( \frac{d}{dx} \sin^{-1}x = \frac{1}{\sqrt{1-x^2}} \)
• \( \frac{d}{dx} \cos^{-1}x = -\frac{1}{\sqrt{1-x^2}} \)

These allow us to deal with more complex functions involving these inverses.

Simplifying expressions in calculus involves reducing the derivative or resulting expression to its most compact form without changing its value. In our current exercise, the derivative we calculated, \( \frac{2\tan^{-1}x}{1+x^2} \), is already in a simplified state. Here's why:

• The expression is free of unnecessary brackets.
• There are no common factors to further reduce it.
• It is neatly presented as a single fraction.

Being comfortable with simplification techniques is crucial as it makes expressions easier to work with, and often more comprehensible. Simplified forms are preferred as they help reveal the function's key characteristics, such as asymptotic behavior and growth rate. However, in some cases, alternative forms might be beneficial if they better match what you are aiming for in an application.