Derivative of a Product of Functions with Trigonometric and Exponential Components

Find $\frac{dW}{dT}$ for $W = x \cdot \sin(y)$ where $x = e^t$ and $y = \pi - t$, and evaluate $\frac{dW}{dT}$ at $t = 0$.

In this problem, we are tasked with finding the derivative of a function that involves a product of trigonometric and exponential components. Specifically, we want to determine how the function W, which is the product of x and the sine of y, changes with respect to T. Here, x and y are themselves functions of T, specifically x equals the exponential function of t, and y is given as pi minus t.

The mathematical concept at play is the chain rule for differentiation, which is crucial when dealing with functions of a function, or composite functions. In particular, we employ the strategy of finding partial derivatives of composite functions with respect to one variable, while understanding how the inner variables themselves depend on that variable. This involves understanding how the rate of change flows through the layers of functions present in this expression. W is a function of x and y, both varied by t, requiring differentiation with respect to these variables first, and then considering the nested dependence on t.

Evaluating this derivative at t equals zero involves substituting this value into our derived expression for the rate of change, ensuring we correctly apply piecemeal each variable's value at this point. This evaluation practice tests our understanding of both the specific derivative calculation and our integrative comprehension of algebraic simplification and substitution within the structure of calculus problems. This problem provides a deep dive into advanced calculus techniques with exponential, trigonometric, and linear adjustments and showcases their applications in real-time calculus scenarios.

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