University:
Washington State University
Course:
MATH 501 | Real Analysis
Academic year:

2023
Views:

441

Pages:

3
Author:

NeatGold

METHODS OF DIFFERENTIATIONDifferntiation by First Principledy = him by = him t (x + Dx) Dx -f(x)dx a-so sx AX-0=f'(x) = y'= = lim f(x+h)- - f(x)hCommon Expressions of Differentiationd(...)d ( )valuevaluedxdx0sintx1/41-x2x"nx"-1co5'x-1/41-x2anallinetain'x1/1+ x2exexcort--1/1+x2sinxcosxsec x1217x2-1CDSx--sinzecosect-1/1x1rx-tanxsec2cotx- cosec2secksecxtanxcosecx- cosecxcotxNote that:d (u+v - w) = du + dv - dwdx:dxdn . dn= ii + v' - w'dxd(k.f(x)) = k df(x) = k f'(x)cxd (u.v) = (uv)' = i'v + v'u (productruledx(ww)' = wive + V uw + w'uvdn vd (u) = vu' - v'u (quotient rule)V2I DatePageDifferentiationofCompositeFunctionsdy =dyde=dxdndydtdt = f'(t) . g'(x)dt dxForexample,dsinx2=dsintwhichdxdxcanbesimplifiedusingthechainrule:dsintxtt=costx2xatdx = Cosx x 2x= 2xwsxxEssentiallywedifferntiatelayerbylayerandmultiplytheresults.DifferenciationofImplicitFunctions2f(x,y)dy =-ox=partialderivativewrtxdxdt(x,4)dypartial deriative wrt ydf? =partialdiff(f)+partialdiff(g)dx=fn48=(NfN-)f'+(A'lnA)g'gfg-1=f9 Int g'DifferentiationofParameticFunctionsandx=h(t),wheretisif y = g(t)a+dydy/dt=parameter,g'(t)axdx/outh'(t) Higher Order DerivativesFor a given function f(x)1st derivativedy/dx = f'(x) = y, = 'y'2nd derivativeday/dx' = f"(x) = yz = y"19thderivativedry/dx" = for = Yu = yn'Note that:driver . day/dy = I but dax/dyz #1day/dx2 = y" = - 2/1x13Nth Derivatives= nlearanareaxprovable usingd" del time1/ax+6(-1)h an not(ax +b)n+1sin(ax+b)an sin (4T/2 + ax + b) same proofcos(ax+b)an cos (40/2 + ax+b)pascal's thianglev) = 6 unv + Guniv, ++us enuvnis(product rule)Differntiation of Determinantst ghIfF(x) =uvwthenF'(x) =l m nf' g' h't g hf g hu V w+w V' w'atu v wl in nl in ul' in n'

Methods of Differentiation

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