3.2 The Derivative as a Function.pdf - I lI ni i Il i 111 I as a function if f X s 2 3 44 find f X cxi limf xn h o fcxsfot.sn h 7 taYadffiffatYoen 6 f

3.2 The Derivative as a Function.pdf - I lI ni i Il i 111 I...

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I l I i ni Il i 111 I as a function if f X s 2 3 44 find f X cxi limf xn fcxsfot.sn taYadffiffatYoen 7 6 46 2 05 y h o h f X 6 2 4 The domain of F is the set of inputs where f is differentiable alternatively ge.iq miaigiiEE g m.hieEiasIni O ii.int m lim 2CxthP 4Cxth 0 h generally requires him zcx73x2ht3xhth3 4x4h 2X3 polynomial long division h O h Hh 28 3 6 2 ht6xh2t2h3 x 4h 2 3 4 1 y T him 6X2ht6Xh2t2h3 4h Mf y f X then y f X y X Z y 3 h 0 h Tiff 61ft Lex h 2h24 I day the derivative of y with respect to X d 2 Theorem expanding from y DICKY Is a theorem involving X fcxt mxtbfkxt.im fy behaves AS an operator d dxM 0
Theorem Power Rule dawn p xn I k l e I l s l If f Cx X 2 then f is mini proof 2 2 1 2 3 f CX X flexi Nim faith f x o h lim h o 4th n xn T lim Mtnxnyntht H h so I lim n xn t h 0 Theorem tinearity Rules suppose f gave differentiable SUMI Difference Rule ft g and f g are differentiable further ft g X FIX gtx and f g CN fix gtx f y suppose fCxkX2 and guys x Power rule f 2x and g CHA SUM Rule ft

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