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Question Number 80442 by Power last updated on 03/Feb/20

Commented by john santu last updated on 03/Feb/20

$$f{}^{\prime }\left(a\right)hahaha$$

Commented by Power last updated on 03/Feb/20

$$\mathrm{solution}\mathrm{sir}\mathrm{pls}$$

Commented by mr W last updated on 03/Feb/20

$$leta=x+\mathrm{\Delta }h$$$$\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$$$$=\underset{x\to a}{\lim }\frac{f\left(a\right)-f\left(x\right)}{a-x}$$$$=\underset{\mathrm{\Delta }h\to 0}{\lim }\frac{f(x+\mathrm{\Delta }h)-f\left(x\right)}{\mathrm{\Delta }h}\Rightarrow definitionof\frac{df\left(x\right)}{dx}$$$$=f^{\prime }\left(x\right)(or=f^{\prime }\left(a\right)insameway)$$

Commented by msup trace by abdo last updated on 03/Feb/20

$$iffisnotderivableatathis$$$$limitdontexistorinfinite...!$$

Answered by mind is power last updated on 03/Feb/20

$$nothingtoosay$$$$iff\left(x\right)=\mid x-a\mid weget\underset{-}{+}1dependinhowx\to a$$$$iffisdifferentiablearrounda$$$$\Rightarrow f\left(x\right)=f\left(a\right)+f^{\prime }\left(a\right)(x-a)+o(x-a)$$$$\Rightarrow \frac{f\left(x\right)-f\left(a\right)}{x-a}=f^{\prime }\left(a\right)+o\left(1\right)$$$$\Rightarrow \underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}=f^{\prime }\left(a\right)$$

Commented by Power last updated on 03/Feb/20

$$\mathrm{thanks}$$

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