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Differentiation is a key concept in calculus, used to determine the rate at which a function is changing at any given point. In simpler terms, it helps us understand how a function behaves as its input values change. To find the tangent line to the function at a particular point, we first need to find its derivative. The derivative, \(f'(x)\), tells us the slope of the tangent line at any point on the function's graph. Mathematically, the derivative of a function \(f(x)\) is defined as: \[f'(x) = \frac{d}{dx}[f(x)]\] This represents the limit of the average rate of change of the function as the interval approaches zero.

The chain rule is a formula used to compute the derivative of the composition of two or more functions. When a function is nested within another function, we use the chain rule to differentiate it. For example, if we have \(h(x) = g(f(x))\), the chain rule states: \[h'(x) = g'(f(x)) \times f'(x)\] In the given exercise, \(f(x) = 2e^{-3x}\) is a composite function. The outer function is \(2u\) and the inner function is \(u = e^{-3x}\). To differentiate this, we first find the derivative of \(2u\) with respect to \(u\), and then multiply it by the derivative of \(u\) with respect to \(x\): \[f'(x) = 2 \times -3e^{-3x} = -6e^{-3x}\] This is how we applied the chain rule to find \(f'(x)\).

The point-slope form is a way to write the equation of a line when you know the slope and a point on the line. The formula is: \[y - y_1 = m(x - x_1)\] Here, \(m\) is the slope of the line, and \( (x_1, y_1) \) is a point on the line. In the context of the given problem, we found the slope of the tangent line by evaluating the derivative at the given point, \(x = 0\). This gave us \(m = -6\). The given point is \((0, 2)\). Substituting these values into the point-slope form equation, we get: \[y - 2 = -6(x - 0)\] Simplifying, this results in: \[y = -6x + 2\] This is the equation of the tangent line to the function at the point \( (0, 2) \).