The total resistance R produced by three conductors with resistances R1, R2, R3 connected in a parallel electrical circuit is given by the formula 1 R

To find $\frac{\partial R}{\partial R_1}$ where the resistance $R$ is given by the formula for three resistors in parallel:

$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

We want to differentiate $R$ with respect to $R_1$. Here's the step-by-step process:

Step 1: Rewrite the formula for $R$

The equation can be rewritten as:

$\frac{1}{R} = f(R_1, R_2, R_3) = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Step 2: Differentiate both sides with respect to $R_1$

We will differentiate both sides of the equation with respect to $R_1$, applying the chain rule:

$\frac{d}{dR_1} \left(\frac{1}{R}\right) = \frac{d}{dR_1} \left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\right)$

Left-hand side:

We know the derivative of $\frac{1}{R}$ with respect to $R_1$ is:

$\frac{d}{dR_1} \left(\frac{1}{R}\right) = -\frac{1}{R^2} \cdot \frac{\partial R}{\partial R_1}$

Right-hand side:

Now, differentiating the right-hand side:

$\frac{d}{dR_1} \left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\right) = -\frac{1}{R_1^2}$

So, we now have:

$-\frac{1}{R^2} \cdot \frac{\partial R}{\partial R_1} = -\frac{1}{R_1^2}$

Step 3: Solve for $\frac{\partial R}{\partial R_1}$

Multiply both sides of the equation by $-R^2$:

$\frac{\partial R}{\partial R_1} = \frac{R^2}{R_1^2}$

Final Answer:

$\frac{\partial R}{\partial R_1} = \frac{R^2}{R_1^2}$

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Would you like further details on this process or a deeper explanation of any of the steps?

Here are 5 related questions you might find interesting:

1. How would you calculate $\frac{\partial R}{\partial R_2}$?
2. What happens to the resistance when $R_1$ becomes very large?
3. How does the total resistance change if the three resistors are identical?
4. What is the formula for the total resistance if there are more than three resistors in parallel?
5. How do you apply this concept to a circuit with resistors in series and parallel combinations?

Tip: Understanding how resistance behaves in parallel is crucial because the total resistance is always less than the smallest individual resistance.