DIFFERENTIABILITYDifferentiation gives up thedirection ofthe graphGeometrically,itistheslopeofthetangent drawn to the curve of a point.Algebraically it is the instantaneousrate of change of f(x) at a point.Right Hand Derivative (R4D) at a point x = ais given by -R4D = f'(a+) = lim Yo-Yphoot xo-xp= lim f(ath) - -fla)hasto(at '))- aLeft Hand Derivative (LHD) at a point x = ais given by -LHD = f'(a-) = h Lim o f(a+h)-Fla) -= lim Yp - YR(a++)-ah -0- x-x -f(x) is differentiable at x = a onlywhen LID = R4D = finiteQ (ath, fla+h))tangent must be nonvertical for t(x)p (a,flas)to be diffableFor f'(a) to be meaningfull, it must be defined at x = aA can be function to determine willusedgraph Iwhether or notabeForornotexamplet(x) differentiable is non diffable it -graph was a corner(LHD E RHD)graph is discontinuousRHD= valid(L4D # RHD)LHD = +00not finitewit - for+s).- agraph has a vertical tangent(LHD oc)note Diff. ability implies cont vity, anddiscout uity implies NON diffabilityalso If at x = a LHD = RHD = finite,f is cont and diff at x = a PageIf x = a . at least one of LHD or R4D= I 00, then f is non diffable forsure, and is discount if no verticaltangent is present at x =aandIff(x) and g(x) are diffable,f + g, f.g and flg are also diffableIff(x) is diftable and n (x) isnon diffable, f + h = non diffableat thesame points and f x h cannotsay When two non diffable funcsareadded we cannot say eithernoteNote that -this rule is only for moduluszero difffinite non diff = diffnon Jero diffnon diff = won diff