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The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. When you have a function nested inside another function, the Chain Rule provides a method to find the derivative. To put it simply, if you have two functions, say an outer function \( f \) and an inner function \( g \), the Chain Rule states that the derivative of their composition, \( f(g(x)) \), is:

• First, differentiate the outer function with respect to the inner function.
• Then, multiply by the derivative of the inner function with respect to \( x \).

This approach ensures you correctly account for the rate changes in both functions. In our example, we have \( y = 3 \tan(3x + 2) \), where \( 3 \tan(u) \) is the outer function and \( u = 3x + 2 \) is the inner function. Applying the Chain Rule effectively helps us determine the overall change in \( y \) as \( x \) changes.

The derivative of the tangent function is a vital concept in calculus. The basic rule for differentiating \( \tan(u) \) is that the result is \( \sec^2(u) \). When the tangent function involves a constant multiple, like \( 3 \tan(u) \), the derivative becomes \( 3 \sec^2(u) \).

Remember:

• \( \tan(u) \) is a trigonometric function related to sine and cosine.
• The derivative \( \sec^2(u) \) indicates the steepness or slope of the curve at any point \( u \).
• \( \sec(u) \) is 1/cosine, making \( \sec^2(u) \) the square of this reciprocal.

In our original equation, we found \( d/du(3 \tan(u)) = 3 \sec^2(u) \). Understanding this rule is crucial for correctly applying it in more complex differentiation problems.

A systematic approach to differentiation can help break down complex problems. Following ordered steps ensures you capture all necessary aspects:

1. **Identify Functions**: Recognize both the outer and inner functions. In the exercise, the outer function is \( 3 \tan(u) \) and the inner is \( u = 3x + 2 \).
2. **Apply the Chain Rule**: To get \( \frac{dy}{dx} \), calculate \( f'(g(x)) \cdot g'(x) \).
3. **Differentiate Outer Function**: Here, \( \frac{d}{du}(3 \tan(u)) = 3 \sec^2(u) \).
4. **Differentiate Inner Function**: The derivative of \( u = 3x + 2 \) is \( \frac{du}{dx} = 3 \).
5. **Combine**: Multiply the derivatives: \( 3 \sec^2(3x + 2) \cdot 3 = 9 \sec^2(3x + 2) \).

Such steps not only simplify the work but also enhance understanding and accuracy.