My professor asked to take the derivative of profit for fun so I just wanted to check if my answer was right.

profit=PQ(p,a)-c(q(a))-a

dΠ/dQ = -2bQ + (a-c) ≡ 0.
Thus, -2bQ = -(a-c) and Q = (a-c)/2b.

Its kinda hard to tell, but something in your answer looks odd. There is nothing in your profit equation that would produce the term "-2bQ"

do you have any pointers on how i could fix this or maybe what is the right way of taking the derivative??

To verify if your answer is correct, let's go through the steps of finding the derivative of the profit equation.

Given: profit = PQ(p, a) - c(q(a)) - a

To find dΠ/dQ, we will differentiate the profit equation with respect to Q while treating all other variables as constants.

1. Differentiate the first term PQ(p, a) with respect to Q:
d/dQ [PQ(p, a)] = P * dQ/dQ * (p, a)
= P * (p, a)
= P * p

2. Differentiate the second term c(q(a)) with respect to Q:
d/dQ [c(q(a))] = c * d(q(a))/dQ

Since q(a) doesn't directly depend on Q, its derivative with respect to Q is zero.
d/dQ [c(q(a))] = c * 0
= 0

3. Differentiate the third term -a with respect to Q:
d/dQ [-a] = - 0 (since a is a constant w.r.t Q)
= 0

Now, let's combine the results:

dΠ/dQ = P * p + 0 - 0
= P * p

Comparing this with the expression you provided:

dΠ/dQ = -2bQ + (a-c)

We can see that our result is not the same as your answer. Therefore, the derivative you calculated is not correct.

To find the correct derivative, let's go through the steps again:

1. Start with the equation: -2bQ + (a-c) = 0

2. Solve for Q by isolating the variable:
-2bQ = -(a-c)
Divide both sides by -2b: Q = (a-c)/(-2b)
Simplify: Q = (c-a)/(2b)

So, the correct derivative of the profit equation with respect to Q is:
dΠ/dQ = P * p

And the correct value of Q is Q = (c-a)/(2b).