Partial Fractions to solve Logistic Equation

Question

I am not really understanding how my book is getting

$$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x}+\frac{x'}{K-x}$$

so $$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x-\frac{x^2}{K}}$$ $$=\frac{\frac{x'}{1}}{\frac{xK-x^2}{K}}$$ $$=\frac{x'K}{xK-x^2}$$ $$=\frac{x'K}{x(K-x)}$$

The book said its "applying partial fractions", but I am confused about how they reached that answer

Answer 1

you can do smething like this $x'\left(\frac{1}{x}+\frac{1}{K-x}\right)=x'\left(\frac{K-x+x}{x(K-x)}\right)$