Question

If $f\left(x\right)=$ $1+x+x^2+x^3+$ $+\dots +x^{99}+x^{100}$ $Find{f}^{\prime }\left(1\right)$

Answer 1

$f\left(x\right)=1+x+x^2+\cdots +x^{99}+x^{100}{f}^{\prime }\left(x\right)=1+2x+3x^2+\cdots +99x^{98}+100x^{99}{f}^{\prime }\left(1\right)=1+2\left(1\right)+3(1{)}^2+\cdots 99(1{)}^{98}+100\left(1{)}^{99}{f}^{\prime }\right(1)=1+2+3+\cdots +99+100=\frac{100\times 101}{2}=5050$