The difference formula is essentially the formula for the slope of a line--more specifically, the secant of another graph. Because of this, we can relate it to the original slope formula, which is (y2-y1)/(x2-x1). Our first step for deriving the difference formula is to find the coordinates of a line that we can apply it to. The graph drawn below illustrates a line and two points on it, which are (x, f(x)) and (x+h, f(x+h)). We use h because it represents the change in x from one point on the x or y axis to another. This is why h is also sometimes known as delta x.
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We want to find the slope of this line, so we plug in the two coordinates into the slope formula. After doing this, we are left with: f(x+h)-f(x) over (x+h)-x. This is already looking like the different quotient we know of, and after this we only have to do a little bit of simplifying. The positive x and the negative x will cancel each other out, and so our final answer would be: f(x+h)-f(x)/h. This is how we would derive the difference quotient from the slope formula.
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