Question

Find the derivative f 0 m(x) of the following function with respect to x: fm(x) = Xm n=1 n x · x n !2

Answer 1

$$f_m(x)=\big(\displaystyle\sum_{ n=1}^m(n^x\cdot x^n)\big)^2$$

$$f_m'(x)=\bigg(\big(\displaystyle\sum_{n=1}^m(n^x\cdot x^n)\big)^2\bigg)'=$$

$$=2\cdot\displaystyle\sum_{n=1}^m(n^x\cdot x^n)\cdot\displaystyle\sum_{n=1}^m\big(\ln(n)\cdot n^x\cdot x^n+n^{x+1}\cdot x^{n-1}\big)$$