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The chain rule is a fundamental technique in calculus for finding the derivative of a composite function. A composite function is essentially a function inside another function. For instance, in our original exercise, we have the function \(f(x) = (\tan x)^{x-1}\), where the inner function is \(u = \tan x\) and the outer function is \(g(u) = u^{x-1}\). The chain rule states that the derivative of a composite function \(f\) is found by multiplying the derivative of the outer function by the derivative of the inner function.

In applying the chain rule, follow these steps:

• Identify the outer and inner functions. For our example, \(u = \tan x\) and \(g(u) = u^{x-1}\).
• Find the derivative of the outer function with respect to the inner function, which gives us \(g'(u) = (x-1)u^{x-2}\).
• Determine the derivative of the inner function with respect to \(x\), yielding \(\frac{du}{dx} = \sec^2 x\).
• Multiply these derivatives: \(f'(x) = g'(u) \cdot \frac{du}{dx}\).

By connecting these dots, we effectively combine differentials to find the derivative of any layered function using the chain rule.

Evaluating derivatives involves not just finding the derivative of a function, but also determining its value at a specific point. In our specific exercise, we computed the derivative of \(f(x) = (\tan x)^{x-1}\) and then evaluated it at \(x = \frac{\pi}{4}\).

Here's how to evaluate derivatives at a given point:

• Start by differentiating the given function. We've already found that \(f'(x) = (x-1)(\tan x)^{x-2}(\sec^2 x)\).
• Substitute the specific value into the derivative. In this case, substitute \(x = \frac{\pi}{4}\) into \(f'(x)\).
• Calculate \(\tan\left(\frac{\pi}{4}\right) = 1\) and \(\sec^2\left(\frac{\pi}{4}\right) = 2\). This stems from knowing the trigonometric values at this angle.
• Simplify the expression to find: \(f'\left(\frac{\pi}{4}\right) = \frac{\pi - 4}{2}\).

Evaluating at a specific point is crucial for solving real-world problems where the instant rate of change is required.

Trigonometric functions like \(\tan x\) and \(\sec x\) play a vital role in calculus, specifically when we're working with functions involving angles. In our exercise, recognizing the properties of these functions is essential.

Here's a brief overview of the trigonometric aspects:

• \(\tan x\), or tangent of \(x\), is the ratio of the opposite side to the adjacent side in a right triangle, based on the angle \(x\).
• The derivative of \(\tan x\) with respect to \(x\) is \(\sec^2 x\), where \(\sec x\), the secant of \(x\), is the reciprocal of cosine.
• Knowing key trigonometric values, such as \(\tan\left(\frac{\pi}{4}\right) = 1\) and \(\sec^2\left(\frac{\pi}{4}\right) = 2\), helps in evaluating derivatives efficiently.

Understanding these functions and their derivatives allows you to handle more complex problems efficiently and predicts how quantities change with angles.