calculus - Evaluate $sum_{n=1}^infty nx^{n-1}$

How can you evaluate $\sum_{n=1}^\infty nx^{n-1} = \frac{1}{(1-x)^2}$ without relying on the fact that it's the derivative of $\sum_{n=1}^\infty x^n = \frac{1}{1-x}$?

Answer

$$a=1+2x+3x^2+4x^3+5x^4+6x^5+...\\|x|<1\\$$ multiply a by x $$xa=x+2x^2+3x^3+4x^4+5x^5+6x^6+...$$ now subtract a and ax $$a-xa=1+(2x-x)+(3x^2-2x^2)+(4x^3-3x^3)+...\\a(1-x)=1+x+x^2+x^3+x^4+x^5+...\\ $$ $$a(1-x)=\frac{1}{1-x}\\a=\frac{1}{(1-x)^2}$$