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Question Number 134884 by abdullahquwatan last updated on 08/Mar/21

$$\mathrm{if}{2\mathrm{x}}^3-2={\int }_a^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt},\mathrm{then}\mathrm{f}{}^{\prime }\left(a\right)=...$$

Answered by bemath last updated on 08/Mar/21

$$\frac{\mathrm{d}}{\mathrm{dx}}[{2\mathrm{x}}^3-2]=\mathrm{f}\left(\mathrm{x}\right)$$$$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)={6\mathrm{x}}^2;\mathrm{f}{}^{\prime }\left(\mathrm{x}\right)=12\mathrm{x}$$$$\Rightarrow \mathrm{f}{}^{\prime }\left(a\right)=12a$$

Commented by abdullahquwatan last updated on 08/Mar/21

$$\mathrm{thx}\mathrm{sir}$$

Commented by bemath last updated on 08/Mar/21

$$\mathrm{you}\mathrm{put}a=1$$

Answered by Ñï= last updated on 08/Mar/21

$$f\left(x\right)=6x^2$$$$f{\left(x\right)}^{\prime }=12x$$$${\int }_a^{x}f\left(t\right)dt={\int }_a^{x}6x^{2}dx=2x^3-2a^3=2x^3-2$$$$\Rightarrow a^3=1$$$$\Rightarrow f{\left(a\right)}^{\prime }=e^{2ik\pi /3}(k=0,1,2)$$

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