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Question Number 95643 by Ar Brandon last updated on 26/May/20

$$\mathrm{Show}\mathrm{that}\mathrm{the}\mathrm{function}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3\mathrm{is}\mathrm{derivable}\mathrm{at}\mathrm{all}$$$$\mathrm{points}{\mathrm{x}}_0\in \mathbb{R}\mathrm{and}\mathrm{that}{\mathrm{f}}^{\prime }\left({\mathrm{x}}_0\right)={3\mathrm{x}}_0^2$$

Answered by Rio Michael last updated on 26/May/20

$$\mathrm{lets}\mathrm{choose}\mathrm{a}\mathrm{general}\mathrm{point}c\in \mathbb{R},$$$$\mathrm{if}f\left(x\right)=x^3\mathrm{is}\mathrm{derivable}\mathrm{or}\mathrm{differentiable}\mathrm{at}x=x_0=c\mathrm{then}$$$$\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}\mathrm{must}\mathrm{exist}.$$$$\mathrm{but}f\left(x\right)=x^3\Rightarrow f\left(c\right)=c^3$$$$\underset{x\to c}{\lim }\frac{x^3-c^3}{x-c}=\underset{x\to c}{\lim }\frac{(x-c)(x^2+2cx+c^3)}{(x-c)}=\underset{x\to c}{\lim }(x^2+2cx+c^2)=3c^2$$$$$$

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