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Parametric equations are a way to represent a curve by expressing the coordinates of the points on the curve as functions of a variable, often denoted by \( t \). This method allows us to describe complex curves that might not be easily represented by a single function. In the given exercise, the curve is represented by the parametric equations \( x = e^{-t} \cos t \) and \( y = e^{-t} \sin t \). The parameter \( t \) varies between \( 0 \) and \( \frac{\pi}{2} \). This approach is particularly useful when dealing with curves that need both sine and cosine components to be expressed accurately. Understanding parametric equations is essential in many areas, such as motion along a path or the study of mechanical linkages.

Derivatives of parametric equations involve finding the rate at which the coordinates \( x \) and \( y \) change with respect to the parameter \( t \). To find the arc length, we first need to compute these derivatives: \( x'(t) \) and \( y'(t) \). This step involves using differentiation rules tailored for product and composite functions. For \( x(t) = e^{-t} \cos t \), the derivative \( x'(t) \) can be computed using the product and chain rules, resulting in \( x'(t) = - e^{-t} \cos t - e^{-t} \sin t \). Similarly, for \( y(t) = e^{-t} \sin t \), the derivative is \( y'(t) = - e^{-t} \sin t + e^{-t} \cos t \). These derivatives determine how steep or flat the curve will be at any given point.

Integral calculus allows us to calculate quantities like area, volume, and in this case, the length of a curve. To find the arc length of a curve expressed in parametric form, we use the formula \[ L = \int_a^b \sqrt{x'(t)^2 + y'(t)^2} \ dt \]. This formula accounts for the infinitesimal adjustments along the curve, summing these small lengths into the total arc length. In this problem, we integrate from \( t = 0 \) to \( t = \frac{\pi}{2} \), achieving our calculation by simplifying the expression \( \sqrt{x'(t)^2 + y'(t)^2} \) into a more manageable form, and then finding the integral of \( e^{-t} \). Evaluating this integral gives us the arc length of the curve over the specified interval.

The product rule is a key differentiation technique used when finding derivatives of products of functions. When you have two functions multiplied together, such as \( u(t) = e^{-t} \) and \( v(t) = \cos t \) or \( \sin t \), the product rule states that the derivative \( (uv)' \) is \( u'v + uv' \). This rule is critical when working with parametric equations, as these often involve products like those seen in our example. By applying the product rule, we find the derivatives \( x'(t) = - e^{-t} \cos t - e^{-t} \sin t \) and \( y'(t) = - e^{-t} \sin t + e^{-t} \cos t \), capturing how both exponential decay and trigonometric oscillation contribute to the curve's shape. Mastering the product rule helps in tackling a wide range of calculus problems.