Help in finding error in derivative quotient rule

Question

So, I have $f(x)=u(x)v(x) \implies f'(x)=u(x)v'(x)+v(x)u'(x)$, and also have $f(x)=\frac{1}{x} \implies f'(x)=-\frac{1}{x^2}$.

I cannot see what is so then wrong with $f(x)=\frac{u(x)}{v(x)} \implies f'(x)=-\frac{u(x)}{v(x)^2}+\frac{u'(x)}{v(x)}$, like after $w(x)=\frac{1}{v(x)}$. Please help.

Answer 1

You forgot a chain rule there. $$\begin{align}\left(\frac{u(x)}{v(x)}\right)' &= \left(u(x)\frac{1}{v(x)}\right)' \\ &= u'(x) \frac{1}{v(x)} + u(x) \left(-\frac{\color{red}{v'(x)}}{v(x)^2}\right) \\ &= \frac{u'(x)}{v(x)} - \frac{u(x)v'(x)}{v(x)^2} \\ &= \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2},\end{align}$$as wanted.