General chain rule (mulitvariable calculus)

[Langrange]

Homework Statement

question from early transcendentals (Edwards, penny) . chapter 12 partial differentiation
problem 38

1/R = 1/R₁ + 1/R₂ + 1/R₃
R is resistance measured in Ω

R₁ and R₂ are 100Ω and increasing with 1Ω/s
R₃ are 200Ω and decreasing with 2Ω/s

Is R increasing or decreasing at that instant ? at what rate ?

Homework Equations

The Attempt at a Solution

so what i have done is this

R(R,R,R) = (1/R₁ + 1/R₂ + 1/R₃)^-1

R₁ = r₁ + t
R₂ = r₂ + t
R₃ = r₃ - 2t

r is the initial values

using the chain rule
dR/dt = dR/dR₁ * dR₁/dt + dR/dR₂ * dR₂/dt + dR/dR₃ * dR₃/dt

i have probably misunderstood something important here . i get some answers but its not right, can someone explain to me how this works and where I am wrong.


the solution is supposed to be 0.24Ω/s

 

[LCKurtz]

Hard to tell where you went wrong without seeing more details. Your work would be easier if you take the equation$$
R^{-1}= R_1^{-1}+R_2^{-1} + R_3^{-1}$$and differentiate that just as it stands implicitly with respect to t. Then put in the values.

 

[Curious3141]

Homework Statement

question from early transcendentals (Edwards, penny) . chapter 12 partial differentiation
problem 38

1/R = 1/R₁ + 1/R₂ + 1/R₃
R is resistance measured in Ω

R₁ and R₂ are 100Ω and increasing with 1Ω/s
R₃ are 200Ω and decreasing with 2Ω/s

Is R increasing or decreasing at that instant ? at what rate ?

Homework Equations

The Attempt at a Solution

so what i have done is this

R(R,R,R) = (1/R₁ + 1/R₂ + 1/R₃)^-1

R₁ = r₁ + t
R₂ = r₂ + t
R₃ = r₃ - 2t

r is the initial values

using the chain rule
dR/dt = dR/dR₁ * dR₁/dt + dR/dR₂ * dR₂/dt + dR/dR₃ * dR₃/dt

i have probably misunderstood something important here . i get some answers but its not right, can someone explain to me how this works and where I am wrong. the solution is supposed to be 0.24Ω/s

Homework Statement

Homework Equations

The Attempt at a Solution

Let $f(R) = \frac{1}{R}$

Now $\frac{df(R)}{dt} = f'(R).\frac{dR}{dt}$

$f'(R)$ is trivial

Now $\frac{df(R)}{dt} = \frac{\partial f(R)}{\partial R_1}\frac{dR_1}{dt}+\frac{\partial f(R)}{\partial R_2}\frac{dR_2}{dt}+\frac{\partial f(R)}{\partial R_3}\frac{dR_3}{dt}$

Can you proceed from there?

 

[Curious3141]

BTW, assuming a linear relationship of the three resistances over time is invalid. All you know (or need to know) is that *at that instant*, the rates of change are as given. At another instant, they may be different, and all the resistances may behave in wildly nonlinear fashion.

 

[Langrange]

but example f(R)/dR₁

how do u do that one. doesn't that require even more substitutions.

because f(R)=1/R

 

[Curious3141]

but example f(R)/dR₁

how do u do that one. doesn't that require even more substitutions.

because f(R)=1/R

Since I've defined $f(R) = \frac{1}{R}$ and since $\frac{1}{R} = \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}$, you simply have $f(R)=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}$. You should be able to work out the partial derivatives very easily from that, and if you can't, I suggest a more thorough review of the topic.