Louisiana State University
Section 3.1 Introducing the DerivativeTopic 1: Tangent Lines and Rates of ChangeDefinition Rate of Change and the Slope of the Tangent LineThe average rate of change of f on the interval [a,x] is the slope of the corresponding secant line.msec=fx-fax-a The instantaneous rate of change of f at a is the slope of the line tangent to the graph of f at the point (a,fa), provided the limit exists.mtan=limx→afx-fax-a Alternate Rate of Change and the Slope of the Tangent LineDefinition The average rate of change of f on the interval [a,a+?x] is the slope of thecorresponding secant line.msec=fa+?x-fa?x The instantaneous rate of change of f at a is the slope of the line tangent to the graph of f at the point (a,fa), provided the limit exists.mtan=lim?x→0fa+?x-fa?x Topic 2: The Derivative FunctionThe slope of the lines tangent to the graph of a function (or the instantaneous rate of change of the function) is called the derivative of the function. For convenience, we will let h represent ?x when defining the derivative.Definition The Derivative FunctionThe derivative of f is the functionf'x=limh→0fx+h-fxh , provided the limit exists and x is in the domain of f. If f'x exists, we say that f is differentiable at x. If f is differentiable at every point of an open interval I, we say that f is differentiable on ic 3: Derivative NotationFor historical and practical reasons, several notations for the derivative are used.right6183630y' f'x dydx ddx(fx) ................
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