I working on differential problems for calculus. One problem asks to estimate max error on the total resistance of three parallel resistors, but I'm a bit stuck on how to take partial derivatives of the total resistance when it is itself expressed as a reciprocal.
$$ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$
How do you deal with the $R$ being expressed like this when taking the partial derivative?
I would have thought something like this:
$$ \frac{\partial{R}}{\partial{R_1}} = \frac{\frac{-1}{R^2}}{\frac{-1}{R_1^2}} = \frac{R_1^2}{R} $$
But this appears to actually be the inverse of the partial.
Best Answer
If you take differentials, $$ -\frac{dR}{R^2}=-\frac{dR_1}{R_1^2}-\frac{dR_2}{R_2^2}-\frac{dR_3}{R_3^2} $$ Now, to find the partial of $R$ w.r.t. $R_1$ you take $dR_2=dR_3=0$ to find $$ \frac{\partial R}{\partial R_1}=\frac{R^2}{R_1^2}. $$