Please see image. This is a screenshot of a lecture slide from a Control Engineering module, however I can't seem to understand how the partial dc/dg was used to give the RHS of the equation in the red ellipse. I've tried deriving it put I'm sure I'm missing a few steps, please can you aid me in my quest
Many Thanks
$${{\delta C\over C}\over {\delta G\over G}}={G\over C}.{\delta C\over \delta G}={G\over {GK\over 1+GK}}.{\delta \over \delta G}\left({GK\over 1+GK}\right)$$ Got it?
We have $C=\frac{G\cdot K}{1+G\cdot K}$
In general the derivative of a fraction is $\left( \frac{u(G)}{v(G)}\right)'=\frac{u'(G)\cdot v(G)-v'(G)\cdot u(G)}{(v(G))^2}$
$u(G)=G\cdot K \Rightarrow u'(G)=\frac{\partial u}{\partial G}=K$
$K$ is a constant due the partial derivative of $u$ w.r.t $G$.
$v(G)=1+G\cdot K \Rightarrow v'(G)=\frac{\partial v}{\partial G}=K$
Therefore
$\frac{\partial C}{\partial G}=\frac{K\cdot (1+G \cdot K)-K\cdot GK}{(1+G\cdot K )^2}$
And $\frac{G}{C}=\Large{\frac{G}{\frac{GK}{1+GK}}}\normalsize{=G\cdot \frac{1+GK}{GK}=(1+GK)\cdot \frac{G\cdot}{GK}}$
