If <em>PV</em> = 800, then <em>P</em> can be written as a function of <em>V</em>,
<em>P(V)</em> = 800 / <em>V</em>
(a) The average rate of change of <em>P</em> as <em>V</em> increases from 200 to 250 in³ is then
(<em>P</em> (250) - <em>P</em> (200)) / (250 in³ - 200 in³) = (3.2 lb/in² - 4lb/in²) / (50 in³)
... = -0.016 (lb/in²)/in³
(Or -0.016 lb/in⁵, but I figure writing the rate as (units of pressure) per (unit volume) makes more sense.)
(b) We can also write <em>V</em> as a function of <em>P</em> :
<em>V(P)</em> = 800 / <em>P</em>
Take the derivative:
<em>V'(P)</em> = - 800 / <em>P</em>²
which immediately demonstrates that <em>V'(P)</em> ∝ 1 / <em>P</em>², as required. (The fish-looking symbol, ∝, means "is proportional to".)
If differentiating is supposed to be more involved, you can use the limit definition:
