Photo-credits: lovely, hand-drawn pictures by Victoria
Looking above at the image, we notice that our first points the x-value is x and our y-value is f(x). Technically looking at the graph not a certain distance is given so that's the reason why we use f(x). For the very same reason the next set of points x-value is x+h and the y-value is f(x+h). The sole purpose of this graph is to find the slope tangent line on the two specific points. To find the slope of the tangent line we must use the slope formula which is m=(y2-y1)/(x2-x1). We plug in the points to the formula and continue on simplifying as much as we can: cancelling/combining like terms, simplifying, etc. Once we have finished the whole lovely process we get the difference quotient! Voila! The difference quotient is extremely helpful in finding derivatives for a certain graph. A visual is displayed below for the process of plugging in to the points to the slope formula ---> to becoming the difference quotient.
Photo-credits: lovely, hand-drawn pictures by Victoria
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