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The Chain Rule is a critical tool in calculus for differentiating composite functions. A composite function is one where you have a function inside another function, like our example with a function of the form \( r = 6(u)^{3/2} \). To find the derivative, the Chain Rule helps us by allowing differentiation in steps.
First, we differentiate the outer function: if \( r = 6u^{3/2} \), we find the derivative with respect to \( u \) as \( \frac{d}{du}[6u^{3/2}] = 9u^{1/2} \).
Next, we differentiate the inner function \( u = \sec \theta - \tan \theta \) with respect to \( \theta \). This step-by-step approach simplifies the process of finding derivatives of more complicated functions.
In our example, after finding these individual derivatives, we multiply them to find the derivative of the composite function with respect to \( \theta \). This two-step differentiation is what the Chain Rule helps to manage efficiently.

Trigonometric functions are fundamental in calculus, especially when dealing with differentiation. In the problem, we encounter \( \sec \theta \) and \( \tan \theta \), which are lesser-known compared to sine and cosine but equally crucial.
Here's a quick rundown on them:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

These definitions are key in knowing how these functions behave and change.
For differentiation:

• The derivative of \( \sec \theta \) is \( \sec \theta \tan \theta \)
• The derivative of \( \tan \theta \) is \( \sec^2 \theta \)

When you put these formulas together, you can successfully differentiate the composite trigonometric function we had, \( \sec \theta - \tan \theta \), in the exercise.

Finding the derivative of composite functions involves combining the derivative of the outer function with the derivative of the inner function.
In our example, we first saw \( r = 6(\sec \theta - \tan \theta)^{3/2} \), which required us to handle dependencies in the functions inside the parentheses.
We performed the following:

• Differentiate the outside: \( \frac{d}{du}[6u^{3/2}] = 9u^{1/2} \), where \( u = \sec \theta - \tan \theta \)
• Differentiate the inside: \( \frac{du}{d\theta} = \sec \theta \tan \theta - \sec^2 \theta \)

Finally, we applied the Chain Rule to combine these:
\[ \frac{dr}{d\theta} = 9(\sec \theta - \tan \theta)^{1/2} \cdot (\sec \theta \tan \theta - \sec^2 \theta) \]
This is a powerful method for dealing with complex functions because it systematically breaks down the function into manageable parts before taking the derivative.