First, let's look at the function f(x) = 2/(x - 3). This is a simple rational function.
The derivative of a function measures how the function changes as its input changes. Basically, it's the rate of change or the slope of the function at a certain point.
To find the derivative of our function, we can apply the rule for the derivative of a quotient of two functions. According to this rule, if we have a function Q(x) = g(x) / h(x), its derivative Q'(x) is given by:
Q'(x) = (g'(x) * h(x) - g(x) * h'(x)) / [h(x)]²
In our case, g(x) is 2 (a constant function) and h(x) is x - 3. The derivative of g(x), g'(x), is 0 because the derivative of a constant is always zero. The derivative for h(x), h'(x), is 1 because the derivative of x is 1 and the derivative of -3 (a constant) is 0.
Let's apply the results to the formula:
Q'(x) = (0 * (x - 3) - 2 * 1) / (x - 3)² = -2 / (x - 3)²
So, the derivative of the function f(x) = 2/(x - 3) is f'(x) = -2 / (x - 3)².
What this implies is that for any x value that you choose (except x = 3, where the function is undefined), this expression will give the slope of the function f at that particular point.