Derivative as Limit of Slopes of Secant Lines
- Jason McCullough
How do we find the slope of f(x) = -x^2 + 4x - 2 when x = 1?
The slope of the lines through the points (x,f(x)) and (x+Δx,f(x+Δx)) slowly approaches 2 as Δx goes to 0. So the slope of f(x) at x =1 is the limit of the slopes of these "secant lines" and the limiting line that just touches the graph of y=f(x) is called the tangent line. Note that the tangent line has the same slope as the graph at the point where they touch. The instantaneous slope of the graph of f(x) is called the derivative.