Effects of change in tax on output Suppose the government imposes a tax on each unit of output
produced of $t. What is the firm’s new profit maximizing level of
output?
The question suggests the use of comparative to determine how a change in tax affects a change in output. The firm would operate under perfect competition conditions.
I started to answer it this way.
Before Tax: 
π = revenue - costs 
π = pq - C(q) 
dπ/dq = p - C'(q) 
Set dπ/dq for profit-maximisation 
0 = p - C'(q) 
p = C'(q) 
p = MC
After Tax: 
π = revenue - costs - tax 
π = pq - C(q) -tq
dπ/dq = p - t - C'(q) 
Set dπ/dq for profit-maximisation 
0 = p - t - C'(q) 
p = t + C'(q) 
p = t + MC 
Next I would try to find out how quantity changes with tax t by taking derivates? Any ideas?
you can do this using comparative statics and envelope theorem. The profit function specification was correct given the assumptions, let me just rewrite it as:
$$\pi (q,t) = p q - C(q) - tq$$
However, given the above we also know that the optimal quantity will depend also on a tax rate, so we will have $q^* = q(t)$.
Now by envelope theorem:
$$\frac{\pi^*(t)}{dt} = \pi'_2(q^*(t),t) = - Q^*(t) $$
So the optimal level of output given the tax $t$ will depend inversely  on change in profit when the tax rate changes or  $-\pi'_2(q^*(t),t) = Q^*(t)$.