(2.1.1)--derivative-definition.pdf

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1、The Derivative1 The DerivativeThe derivative of f(x)is designated as f(x)or f or y.hxfhxfxfyh)()(lim)(0-+=If this limit does exist,we say that f is Differentiable(可导)at x.Finding a derivative is Called differentiation(求导).Equivalent forms for the Derivativehcfhcfcxcfxfcfhcx)()(lim)()(lim)(0-+=-=Incr

2、ements0limxx00)()(xxxfxf-xyxDD=D0lim)()(0 xfxfy-=D0 xxx-=DNotation;0 xxy=;)(0 xf;dd0 xxxy=0d)(dxxxxf=That is0 xxy=)(0 xf=xyxDD=D0limIncrement of yIncrement of xDifferentiability implies continuity.If the graph of a function has a tangent at point c,then there is no“jump”on the graph at that point,th

3、us is continuous there.The graph of the derivativeThe derivative f(x)gives the slope of the tangent line to the graph of y=f(x)at the value of x.Slope up,the derivative is positiveSlope down,the derivative is negativeThe DerivativeCalculus Xuefen GaoRules for Finding DerivativesProduct RuleIf f and g are differentiable functions,then)()()()()()(xfxgxgxfxgf+=Quotient RuleLet f&g be differentiable functions with g(x)not equal 0.2)()()()()()(xgxgxfxfxgxgf-=Thank you!