I have been studying introductory calculus for a while and on the side I study chemistry. The other day I was messing around with thermodynamics and so I wanted to find the change in vollume if a air duct changes its radius and length. But I am stuck because I have no idea how to figure that out. I am using the equation: $$V = \pi r^2 h$$ How could I find the derivative of the vollume when the height (h) and radius (r) change?
2026-03-25 14:22:58.1774448578
Process of finding the derivative of a function with respect to two variables
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One way to do it is to assume that $r$ and $h$ are functions of some as of yet unidentified parameter.
And then apply the product rule.
$v' = \pi (2rr') h + \pi r^2 h'$
And then you can decide after the fact how much you want $r$ and $h$ to vary and see the impact on $v'$
You could also write it as: $dv = (\pi 2rh)dr + (\pi r^2) dh$