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Question Number 151884 by mathdanisur last updated on 23/Aug/21

$$\mathrm{f}\left(\mathrm{x}\right)=(\mathrm{x}-1)(\mathrm{x}-2)...(\mathrm{x}-2021)$$$$\mathrm{f}{}^{{}^{\prime }}\left(2021\right)=?$$

Answered by mr W last updated on 24/Aug/21

$$\ln y=\ln (x-1)+\ln (x-2)+...+\ln (x-2021)$$$$\frac{y^{\prime }}{y}=\frac{1}{x-1}+\frac{1}{x-2}+...+\frac{1}{x-2021}$$$$y^{\prime }=(x-1)(x-2)...(x-2021)[\frac{1}{x-1}+\frac{1}{x-2}+...+\frac{1}{x-2021}]$$$$y^{\prime }{\mid }_{x=2021}=2020\times 2019\times ...\times 2\times 1=2020!$$

Commented by mathdanisur last updated on 24/Aug/21

$$\mathrm{Thank}\mathrm{you}\mathbf{S}\mathrm{er}$$

Answered by Olaf_Thorendsen last updated on 23/Aug/21

$$f\left(x\right)=\underset{k=1}{\prod ^{2021}}(x-k)$$$$f\left(x\right)=(x-2021)\underset{k=1}{\prod ^{2020}}(x-k)$$$$\mathrm{Let}g\left(x\right)=\underset{k=1}{\prod ^{2020}}(x-k)$$$$f\left(x\right)=(x-2021)g\left(x\right)$$$$f^{\prime }\left(x\right)=(x-2021)g^{\prime }\left(x\right)+g\left(x\right)$$$$f^{\prime }\left(2021\right)=g\left(2021\right)=\underset{k=1}{\prod ^{2020}}(2021-k)$$$$f^{\prime }\left(2021\right)=\underset{k=1}{\prod ^{2020}}k=2020!$$

Commented by mathdanisur last updated on 23/Aug/21

$$\mathrm{Thank}\mathrm{You}\mathbf{S}\mathrm{er}$$

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