Can someone explain the derivation for this profit function

Question

So, how come the derivative ends up as f'-q/l

Some other identities are qi/q=Li/L, qi=Li/L * f(L) and L=sumLi

enter image description here

Answer 1

At its current state, I'm not sure if it's possible to answer the question, but I can start computing the derivative of the following expression: $$ \begin{split} \frac{d}{dL_i} \left( \frac{p L_i f(L)}{L} \right) &= p\frac{d}{dL_i} \left( \frac{ L_i f(L)}{L} \right) \\ &= p \frac{L \left[ L_i f\right]' - L_if\left[L \right]'}{L^2} \\ &= p \frac{L L_i' f + L L_i f' - L_i f L'}{L^2} \\ &= p \frac{L L_i' + L_i \left(Lf' - f' L\right)}{L^2} \\ &= p \left[ \frac{L_i' f}{L} + \frac{L_i}{L}\left(f' - \frac{fL'}{L} \right) \right] \end{split} $$ ... which is already quite close to the desired answer. The main point in the differentiation was to remember the rule of differentiating $$ \left[\frac{f}{g} \right]' = \frac{gf' - fg'}{g^2} $$ The "dot/prime" of course represents differentiation with respect to $L_i$ in these equations. Can you work out the solution with this?