The Derivative of a Straight Line
What a derivative is, read straight off the slope: constant functions have derivative zero, lines f(x) = cx have derivative c, and the formal limit definition turns a secant into a tangent.
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What a derivative is, read straight off the slope: constant functions have derivative zero, lines f(x) = cx have derivative c, and the formal limit definition turns a secant into a tangent.
The derivative is another function — one that describes how the original changes at every point on its graph. Reading left to right: a positive derivative means the function is increasing, a negative one means it is decreasing, and a derivative of zero marks a horizontal tangent.
The graph of is a horizontal line. Its value never changes as runs from to — with no change at all, its derivative is zero everywhere. The same holds for any constant function.
Now let . Every time increases by 1, the function also increases by 1, so its derivative is 1. Scale it up: increases 10 units for every unit increase in , so its derivative is 10. In general, if , the constant is the slope — and that slope is the derivative.
The same idea runs in reverse for a falling line.
f(x) = −x
f(x) = −10x
A line's slope is obvious because it never changes. A curve's steepness varies from point to point, so we need a general definition. Start with two points on the graph, one at and another at . Their average rate of change is:
As shrinks toward 0, the two points slide together and the line joining them — the secant — approaches the tangent at a single point. That limiting slope is the derivative:
The formula looks complicated, but the meaning is simple: it measures how quickly the function is changing at one exact moment.