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To understand the process of finding the slope of the tangent line to a curve, we first need to grasp the concept of differentiation. Differentiation is the process of finding the derivative of a function. The derivative of a function gives us the rate at which the function's value changes at any given point. For any function of the form y = f(x), the derivative f'(x) tells us how y changes as x changes.

In the context of our exercise, we were given the function y = x e^x and needed to find its derivative. Differentiation helps us find this derivative, which we used to determine the slope of the tangent line at a specific point.

The product rule is a fundamental rule in calculus for finding the derivative of the product of two functions. For functions u(x) and v(x), the product rule is stated as: \[ \frac{d}{dx}(u v) = u' v + u v' \]

This means that to differentiate a product of two functions, we take the derivative of the first function and multiply it by the second function, then add the product of the first function and the derivative of the second. In our problem, we had the function y = x e^x, which is the product of x and e^x.

Applying the product rule meant setting u = x and v = e^x. By differentiating each part (\frac{d}{dx}(x) and \frac{d}{dx}(e^x)), we could construct the expression for the derivative of the product.

Exponential functions are a special class of mathematical functions where the variable appears in the exponent. The general form of an exponential function is y = a e^(bx), where e is the base of natural logarithms (approximately 2.71828).

One of the remarkable properties of the exponential function is that its derivative is itself. In other words, \[ \frac{d}{dx}(e^x) = e^x \]

This property greatly simplifies differentiation when exponential functions are involved. In the exercise, because one part of our original function was e^x, we leveraged this property when applying the product rule. It simplifies calculations and is a key reason why exponential functions are prevalent in mathematical modeling and real-world applications like population growth and radioactive decay.