Question

Find partial derivative dR/dR_1: 1/R = 1/R_1 + 1/R_2 + 1/R_3

Answer 1

\[\Large \frac{1}{R}\quad=\quad \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\]Can be written as,\[\Large R^{-1}\quad=\quad R_1^{-1}+R_2^{-1}+R_3^{-1}\]They want u to find \(\Large \dfrac{\partial R}{\partial R_1}\), yes?

Answer 2

yes

Answer 3

So I guess we would ummmm, treat R_2 and R_3 as constants with respect to R_1, so taking their derivatives will simply give us zero.

As for the R and R_1, we can apply the power rule, followed by the chain rule.
On the left side, power rule gives us,\[\Large -R^{-2}\]But chain rule also tells us to multiply by the derivative of the inner function, R in this case, so we get our derivative term attached to it.\[\Large -R^{-2}\frac{\partial R}{\partial R_1}\]

Answer 4

okay

Answer 5

and then just rearrange terms?

Answer 6

Well we need to take the derivative of the right side as well.

Answer 7

right

Answer 8

Power rule gives us something very similar,\[\Large -R_1^{-2}\]Since we're taking our derivative with respect to `this` variable, R_1, we don't care about the dR_1/dR_1 that would show up, it's insignificant.

Answer 9

So we end up with:\[\Large -R^{-2}\frac{\partial R}{\partial R_1}\quad=\quad -R_1^{-2}\]
And yes, you have the right idea, we can now do some rearranging to solve for the derivative term!

Answer 10

alright, thank you!

Answer 11

np c: