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The chain rule is a powerful tool in calculus used to differentiate composite functions. It helps us find the derivative of a function that is nested inside another function. Imagine you are peeling layers of an onion, that's how the chain rule works, peeling each layer to get to the core.

When using the chain rule, you take the derivative of the outer function with respect to the inner function, and then multiply it by the derivative of the inner function with respect to the variable. It's like following a chain of steps to reach your final destination. For example, consider a function like \[ y = f(g(x)) \]To differentiate this, you would first find the derivative of \( f \) with respect to \( g \), and then multiply it by the derivative of \( g \) with respect to \( x \).

The formula for the chain rule is:
\[ \frac{dy}{dx} = \frac{df}{dg} \times \frac{dg}{dx} \]
In the context of our exercise, when we differentiated \( y = 9 \tan(\frac{x}{3}) \), we let \( u = \frac{x}{3} \). The chain rule then guides us to find the derivative of the tangent function first and multiply it by the derivative of \( u \) with respect to \( x \).

The tangent function is a trigonometric function, which means it has special rules for differentiation. Specifically, the derivative of \( \tan(u) \) is \( \sec^2(u) \). This is crucial for solving problems involving derivatives of trigonometric functions.

The \( \sec^2 \) term comes from the fact that the tangent function can be expressed in terms of sine and cosine. The derivative of the tangent function involves the secant function notably because \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)and differentiating this with respect to \( x \) gives us:\[ \frac{d}{dx}\left(\tan(x)\right) = \frac{d}{dx}\left(\frac{\sin(x)}{\cos(x)}\right) = \sec^2(x) \]
In our example, using \( y = 9 \tan(\frac{x}{3}) \), we differentiated as\:\[ \frac{d}{dx}(\tan(u)) = \sec^2(u) \frac{du}{dx}\]Here, this was applied as part of using the chain rule.

Trigonometric differentiation refers to the differentiation techniques applied specifically to trigonometric functions such as sine, cosine, and tangent. These functions have specific derivative rules that are often used in calculus problems.

Here are some basic derivatives of common trigonometric functions:

• \( \frac{d}{dx} \left( \sin(x) \right) = \cos(x) \)
• \( \frac{d}{dx} \left( \cos(x) \right) = -\sin(x) \)
• \( \frac{d}{dx} \left( \tan(x) \right) = \sec^2(x) \)
• \( \frac{d}{dx} \left( \sec(x) \right) = \sec(x)\tan(x) \)
• \( \frac{d}{dx} \left( \csc(x) \right) = -\csc(x)\cot(x) \)
• \( \frac{d}{dx} \left( \cot(x) \right) = -\csc^2(x) \)

These derivatives are essential building blocks when calculating more complex derivatives involving trigonometric expressions. In the given exercise, we use the derivative of \( \tan(x) \) as \( \sec^2(x) \) as a fundamental tool. With these derivatives in hand, you can break down and solve a wide array of calculus problems involving trigonometric functions.