91Ó°ÊÓ

In calculus, the chain rule is an essential method used to differentiate composite functions. You might wonder, "What is a composite function?" Simply put, it's a function made up of other functions. For instance, when one function is nested inside another, like our original function \[ f(x)= \arctan \left( \frac{x}{a} \right) \] Knowing how to apply the chain rule helps find the derivative of such functions. Here's the basic idea: you need to differentiate the outer function and multiply it by the derivative of the inner function.

To help you visualize this:

• Let the outer function be \( F(u) = \arctan(u) \).
• The inner function is \( u = \frac{x}{a} \).

The chain rule allows us to find \( f'(x) \) by differentiating \( F \) with respect to \( u \) and then \( u \) with respect to \( x \). This means: \[ f'(x) = F'(u) \cdot u' \] This approach ensures you can tackle a wide range of problems in calculus with confidence.

Arctangent, written as \( \arctan(x) \), is one of the inverse trigonometric functions. These functions are crucial in mathematics because they allow us to find angles when their tangent, sine, or cosine values are known.

In this problem, we're focusing on \( \arctan \), which reverses what the tangent function does. Instead of taking an angle and giving its tangent, arctangent takes the tangent of an angle and returns the angle itself. The function's domain is all real numbers, while its range is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Understanding the derivative property here is key. For \( \arctan(u) \), the formula for its derivative is: \[ \frac{d}{dx}(\arctan(u)) = \frac{1}{1 + u^2} \cdot \frac{du}{dx} \] This formula originates from the properties of the tangent and its inverses, and helps when dealing with problems involving arctangent differentiation.

The derivative calculation for our function \( f(x)=\arctan\left(\frac{x}{a}\right) \) involves some straightforward steps given in the solution. Let's break it down further!

Step 1: Identify \( u \), the function inside \( \arctan \). Here, \( u = \frac{x}{a} \). Differentiating \( u \) with respect to \( x \), we get \( u' = \frac{1}{a} \) since \( a \) is a constant.

Step 2: Use the arctangent derivative formula. Substitute \( u \) and \( u' \) into: \[ \frac{1}{1 + (\frac{x}{a})^2} \cdot \frac{1}{a} \] Step 3: Simplify the expression to obtain: \[ \frac{1}{a + \frac{x^2}{a}} \] These calculations show the power of combining the chain rule with trigonometric derivatives. Once mastered, you'll find such problems much simpler and more intuitive!