1. ## Using diferentials

Let there be 3 resistors in parallel. We have that the total resistance R satisfies:

$\displaystyle 1/R=1/R_1+1/R_2+1/R_3$. Find dR.

When I tried to do this, I ended up getting a ridicilously long answer. I was just wondering if someone wouldn't mind helping me out with this. Thanks

2. Originally Posted by HelloWorld2
Let there be 3 resistors in parallel. We have that the total resistance R satisfies:

$\displaystyle 1/R=1/R_1+1/R_2+1/R_3$. Find dR.

When I tried to do this, I ended up getting a ridicilously long answer. I was just wondering if someone wouldn't mind helping me out with this. Thanks
Dear HelloWorld2,

$\displaystyle dR=\frac{\partial R}{\partial R_1}dR_1+\frac{\partial R}{\partial R_2}dR_2+\frac{\partial R}{\partial R_3}dR_3$

Since, $\displaystyle \frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R _3}$

$\displaystyle -\frac{1}{R^2}\frac{\partial R}{\partial R_1}=-\frac{1}{R_{1}^{2}}\Rightarrow{\frac{\partial R}{\partial R_1}=\frac{R^2}{R_{1}^{2}}}$

Similarly, $\displaystyle \frac{\partial R}{\partial R_2}=\frac{R^2}{R_{2}^{2}}}$

$\displaystyle \frac{\partial R}{\partial R_3}=\frac{R^2}{R_{3}^{2}}}$

Therefore, $\displaystyle dR=R^2\left(\frac{dR_1}{R_{1}^{2}}+\frac{dR_2}{R_{ 2}^{2}}+\frac{dR_3}{R_{3}^{2}}\right)$