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Question

In an electric circuit with two resistors of resistance $R_1$ and $R_2$ connected in parallel, the total resistance $R$ is given by the formula

$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$

Show that

$R_1 \frac{\partial R}{\partial R_1}+R_2 \frac{\partial R}{\partial R_2}=R$

Solution

VerifiedAnswered 1 year ago

Answered 1 year ago

Step 1

1 of 3To find the partial derivatives $\frac{\partial R}{\partial R_1}$ and $\frac{\partial R}{\partial R_2}$,we first find the expression of the total resistance $R$ of the given expression:

$\begin{aligned} \frac{1}{R}&=\frac{R_2}{R_1R_2}+\frac{R_1}{R_1R_2}\\ \frac{1}{R}&=\frac{R_1+R_2}{R_1R_2}\\ R&=\frac{R_1R_2}{R_1+R_2}\end{aligned}$

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