Basic Derivatives
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The derivative of a function f(x) is written as f’(x), and read as “f prime of x.” By definition, the derivative is the slope of the original function. Let’s find out why.
The Difference Quotient
Secant and Tangent Lines
A tangent line is a line that juuussst barely touches the edge of the graph, intersecting it at only one specific point. Tangent lines look very graceful and tidy, like the first graph on the left. A secant line, on the other hand, is a line that runs right through the middle of a graph, sometimes hitting it at multiple points, and looks generally meaner, like the second graph on the left. It’s important to realize here that the slope of the secant line is the average rate of change over the interval between the points where the secant line intersects the graph. The slope of the tangent line instead indicates an instantaneous rate of change, or slope, at the single point where it intersects the graph.
Creating the Derivative
If we start with a point (c,f(c)) on a graph, and move a certain distance, delta x, to the right of that point, we can call the new point on the graph (c+delta x,f(c+delta x)). Connecting those points together gives us a secant line, and we can use the slope equation
to determine that the slope of the secant line is
which, when we simplify gives us
I hope your heart just skipped a beat out of pure excitement.
The point of all this nonsense is that, if I take my second point and start moving it slowly left, closer to the original point, the slope of the secant line becomes closer to the slope of the tangent line at the original point.
Running through this exercise allows us to realize that if I reduce delta x to zero and the distance between the two secant points becomes nothing, that the slope of the secant line is now exactly the same as the slope of the tangent line. In fact, we’ve just changed the secant line into the tangent line entirely. That is how we create the formula above, which is the very definition of the derivative, which is why the definition of the derivative is the slope of the function at a single point.
