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The profit function is given by:
$$
\pi = PQ(P) - C(Q(P), \mu) - tQ(P)
$$
Assume demand is linear, s
$$
p = \alpha - \beta Q \to Q = b(\alpha - P),
$$
where $b = 1/\beta$. So:
$$
Q_P = -b,
$$
where I use subscripts to denote the partial derivatives.
The first order condition for profit maximisation gives:
$$
\begin{align*}
&Q + P Q_P - C_Q Q_P - t ...
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