Definition of a Derivative In Calculus, we talk about the rate at which a function is changing at a point. Essentially, we are asked to find the slope of the line that is tangent to a curve at a singlepoint. If you think about this for a moment, something should strike you as wrong. One cannot find a single line with only one point. There are infinitely many lines that pass through one point. Consider the line ymx=for any m, it passes through the origin. We need two points to find the equation of a line. With two points, we can find the slope between the two points. The equation for slope is 2121()yy fx fxyxxxxx−−∆==∆−−. Suppose that we wanted to find the slope of the tangent line at a point x. We said that we need two points. So, let’s pick a second point just slightly to the right of
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