I have the following two functions
$$f(x)=\frac{abc^{r-1}x^rr}{(ax^r+bc^r)^2} , g(x)=\frac{abc^{r}x^{r-1}r}{(ax^r+bc^r)^2}$$
Define $h(x):=f(x)+g(x)$. Given that the derivative of a sum of functions is the sum of derivatives, we know that $h'(x)=0 \iff f'(x)+g'(x)=0$.
Through differentiating I get that
$$h'(x)=0 \iff-\dfrac{abc^{r-1}r^2x^{r-1}\left(ax^r-bc^r\right)}{\left(ax^r+bc^r\right)^3}+-\dfrac{abc^rrx^{r-2}\left(\left(ar+a\right)x^r-bc^rr+bc^r\right)}{\left(ax^r+bc^r\right)^3} =0$$
From a comment, I know that I can multiply this by $\frac{D}{abc^{r-1}x^{r-2}}$, where $D=\left(ax^r+bc^r\right)^3$. This yields
$$h'(x)=0 \iff -rx(ax^r-bc^r)-c((ar+a)x^r-bc^rr+bc^r)) =0$$
$$bc^rrx-arx^{r+1}=acrx^r-acx^r-bc^{r+1}r+bc^{r+1}$$
From which I simply cannot solve a condition for $h'(x)=0$. $x$ has too many different powers. Is there another alternative? Any suggestions for how to specify $x$ such that $h'(x)=0$?