# Cournot's game profit maximisation first order condition question

My question is about the last part (I attached the rest for context). I don't understand the final line at all. I thought the first order condition would just be to set the derivative = 0. Very confused! Any help would be greatly appreciated! Thank you :D

• Hint: Use product rule and rearrange. – Herr K. Apr 16 '19 at 20:39

The product rule of differentiation is $$\frac{d}{dx}fg=f’g+fg’$$ The text in your question states that you are trying to maximize the quantity $$[a-Q-c]q_i$$ So, as you predicted, we must differentiate this and find when the derivative equals $$0$$. If we differentiate this with respect to $$q_i$$ and use the product rule, setting $$f=a-Q-c$$ and $$g=q_i$$, we obtain $$-\frac{dQ}{dq_i}q_i+[a-Q-c]$$ since the derivative of $$a-Q-c$$ with respect to $$q_i$$ is $$\frac{dQ}{dq_i}$$ (since $$a,c$$ are constant) and the derivative of $$q_i$$ w.r.t $$q_i$$ is $$1$$. Setting this equal to zero gives $$-q_i\frac{dQ}{dq_i}+[a-Q-c]=0$$ or, equivalently, $$a-Q-c=q_i \frac{dQ}{dq_i}$$ which is the last line of the text.