A tangent line is just a line that touches the graph once. Here is a graph called f with a point labeled x and we calculate the slope of the tangent line at the point x with the difference quotient. The length from the origin to x is called "x." The coordinates at that point x (that is touching the tangent line) is (x,f(x)). Another point to the right of x is called "h(delta x or change in x)". So this new point has the coordinates of (x+h,f(x+h)). A secant line is a line that goes through two points. To find the slope of the secant line you need to use the point slope formula, which is y2-y1/x2-x1. Then, you just plug in the x and y values and you get f(x+h)-f(x)/ x+h-x. The x's will cancel in the denominator and you are left with f of x plus h minus f of x divided by h that's the difference quotient! So the difference quotient represents the slope of the secant line. The smaller the change in x, the more the secant line resembles the tangent line. To make the change in delta x smaller we use a limit in the secant slope. So we take the limit as h(delta x) approaches 0 because we want our delta x to be as small as possible for the secant line to resemble the tangent line. We can take the limit as h approaches 0 because approaches means getting close, but never getting there and that's exactly what we need (to get close to zero but not zero because we can't have 0 in our denominator). The difference quotient is the definition of the derivative.

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