2.2 Derivative. 1. Definition of Derivative at a Point: The derivative of the function f x at x a is defined as
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1 . Derivative. Definition of Derivative at a Point: Te derivative of te function f at a is defined as f fa fa a lim provided te limit eists. If te limit eists, we sa tat f is differentiable at a, oterwise, we sa tat f is not differentiable at a. An alternative form of f a : f f fa a lim a a provided te limit eists. Remarks: a. From te definition of f a, we know te derivative of f at a is te same as te slope of te tangent line to te curve f at a; and it is also te instantaneous rate of cange of f at a. b. Wen f is a position function of a moving object, f a gives te velocit of te object wen a. Eample Find te derivative of f at. a lim : i. f f f f f i lim lim iv. f b. Use te formula: f a lim a f fa a : i. f f f f f i lim lim iv. f Eample Find te derivative of f at. a lim : i. f f f i lim f f lim
2 iv. f b. Use te formula: f f fa a lim a a : i. f f f i lim f f lim Eample Find te derivative of f at. a lim : i. f f f i lim f f lim iv. f b. Use te formula: f a lim a f fa a : i. f f f i iv. lim f f lim. Definition of te Derivative Function: Te derivative of a function f is te function f defined as f f f lim provided te limit eists. Te process of computing a derivative is called differentiation. Eample Witout computing, find te derivative function f for a. f c b. f m b a. f because te tangent line to te curve c at an point a is te line a wose slope is.
3 b. f m because te tangent line to te curve m b at an point a is te line m b wose slope is m. Eample Find f if it eists were a. f b. f for a. b. c. f f f lim f f f f f lim f f c. f for lim f f f lim lim,for. f f lim,for.. Sketcing te Grap of f from te Given Grap of f : From te definition of f a, we know f a m tan at te point a, fa. So, for a given grap of
4 f we can sketc a roug grap of f b estimate m tan at as man points as possible. Te following are eamples of eact graps of f and f. In class, we will sow ow to sketc roug graps of f from given f f,... f - f,... f f,... f - f,... f Eample Page 7: -6, 7. Alternative Derivative Notations: Te following are all alternative for te derivative function: let f d d is also called a differential operator. f d d df d d d f. Nondifferentiable Points of a Function: From te definition of f a, we know te function is not differentiable at a if te limit f f lim does not eist.
5 Note tat te following are several conditions wit wic f a does not eist: a. f is not continuous at a ( fa is not defined; lim f DNE or lim f fa; f f b. lim ; f f c. lim DNE; On te oter and, f is continuous at a if f a eists. Eample Page f is not differentiable at, and sincef is not continuous at tese. f f f f. f is not differentiable at since lim lim. f is not differentiable at. if Eample Let f. Sow grapicall and algebraicall tat f is continuous if at but f is not differenitable at. Grapicall, we see f is continuous at and is not differentiable at. - - Algebraicall, since lim f lim and lim f lim, f is f f f f continuous at. Now ceck lim and lim. lim f f f f f lim, lim Hence, f is not differentiable at since lim if if f f f f lim lim f f
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