We now give a rigorous definition of the derivative, along the lines of the definition of tangent line given above as a limit of certain secant lines.
A secant line for the function f (x) at x = x _{0} is a line through the points (x _{0}, f (x _{0})) and (x, f (x)) , for some x in the domain of f . The slope of such a secant line is
The derivative of f at x _{0} is the limit of the slopes of the secant lines at x _{0} as x approaches x _{0} (that is, as the secant lines approach the tangent line). Thus we have the following formula for the derivative of f at x _{0} :
f'(x _{0}) = (x _{0}) = |
If we let Δx = x - x _{0} , the change in x , then x = x _{0} + Δx and substitution yields an alternate formula for the derivative:
f’(x _{0})) = (x _{0}) = |
The quotients in the above expressions are often referred to as difference quotients.