How to find partial derivative

Question

I have this equation

$$π =(1000 − y_1 − y_2)(y_1 + y_2) - y_1^2 - 200y_2 -y_2^2$$

which simplifies to

$$1000y_1 + 800y_2 − 2y_2^2 - 2y_1^2 - 2y_12y_2$$

Variables are $y_1$ and $y_2$

It says that the partial derivative with respect to $y_1 =1000 − 2y_2 − 4y_1$ But i cant seem to get this answer ?

Answer 1

Your simplification is wrong, you should not have a term like $2y_12y_2$.

\begin{align} \pi &=(1000-y_1-y_2)(y_1+y_2)-y_1^2-200y_2-y_2^2\\ &=1000y_1+1000y_2-y_1^2-y_1y_2-y_2y_1-y_2^2-y_1^2-200y_2-y_2^2\\ &=1000y_1+800y_2-2y_1^2-2y_1y_2-2y_2^2 \end{align}

Now take the partial derivative and you will get the answer.