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Explain-the-proof-with-appropriate-diagram-Lim-h-0-f-x-f-x-h-h-dy-dx-where-y-f-x-




Question Number 75402 by vishalbhardwaj last updated on 10/Dec/19
Explain the proof   with appropriate  diagram :   Lim_(h→0) ((f(x)−f(x−h))/(−h))      = (dy/dx) , where y = f(x)
$$\mathrm{Explain}\:\mathrm{the}\:\mathrm{proof}\: \\ $$$$\mathrm{with}\:\mathrm{appropriate} \\ $$$$\mathrm{diagram}\::\: \\ $$$$\mathrm{Lim}_{{h}\rightarrow\mathrm{0}} \frac{{f}\left({x}\right)−{f}\left({x}−{h}\right)}{−{h}}\: \\ $$$$\:\:\:=\:\frac{{dy}}{{dx}}\:,\:\mathrm{where}\:{y}\:=\:{f}\left({x}\right) \\ $$
Commented by Kunal12588 last updated on 10/Dec/19
isn′t it the defination of derivative?  first principle  (dy/dx)=lim_(Δx→0) ((Δy)/(Δx))=lim_(Δx→0) ((f(x+Δx)−f(x))/(Δx))
$${isn}'{t}\:{it}\:{the}\:{defination}\:{of}\:{derivative}? \\ $$$${first}\:{principle} \\ $$$$\frac{{dy}}{{dx}}=\underset{\Delta{x}\rightarrow\mathrm{0}} {{lim}}\frac{\Delta{y}}{\Delta{x}}=\underset{\Delta{x}\rightarrow\mathrm{0}} {{lim}}\frac{{f}\left({x}+\Delta{x}\right)−{f}\left({x}\right)}{\Delta{x}} \\ $$
Commented by vishalbhardwaj last updated on 10/Dec/19
sir  this is LHD
$$\mathrm{sir}\:\:\mathrm{this}\:\mathrm{is}\:\mathrm{LHD} \\ $$
Commented by Kunal12588 last updated on 10/Dec/19
Δx=0−h  ⇒Δx=−h  Δx→0⇒h→0  LHD=lim_(h→0)  ((f(x−h)−f(x))/(−h))
$$\Delta{x}=\mathrm{0}−{h} \\ $$$$\Rightarrow\Delta{x}=−{h} \\ $$$$\Delta{x}\rightarrow\mathrm{0}\Rightarrow{h}\rightarrow\mathrm{0} \\ $$$${LHD}=\underset{{h}\rightarrow\mathrm{0}} {{lim}}\:\frac{{f}\left({x}−{h}\right)−{f}\left({x}\right)}{−{h}} \\ $$
Commented by vishalbhardwaj last updated on 12/Dec/19
please explain the proof of this  with the help of diagram
$$\mathrm{please}\:\mathrm{explain}\:\mathrm{the}\:\mathrm{proof}\:\mathrm{of}\:\mathrm{this} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{help}\:\mathrm{of}\:\mathrm{diagram} \\ $$

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