$R = dV/dI$ for varying temperature

Question

I'm trying to do my prelab for an E&M course, and am asked if, for plotting $V$ vs $I$ with a varying temperature, I should expect a linear slope. I know that both $V$ and $I$ depend on $R$, and since $R$ is proportional to $T$ that I shouldn't.

Trying to show this, I'm having some problems -- probably because my differential equations skills are a bit rusty.

Simplely, if $V = I \,R(T)$, how do I take $\mathrm{d}V/\mathrm{d}I$?

I can find $\mathrm{d}V/\mathrm{d}T$ by chain rule, but i'm just drawing a blank with $\mathrm{d}V/\mathrm{d}I$ since $I = V/R(T)$

Answer 1

The missing piece here is that the temperature of the resistor is a function of the current. Your equation should perhaps read $V = I\,R(T(I))$. Does that help?

Answer 2

You have a function: $V(T, i) = i \cdot R(T)$ and you should get $\dfrac{dV}{di}$.

$T$ doesn't change when you vary $i$ and $R(T)$ doesn't too, so it can be considered as a constant comparing to variable $i$.

Fix $T$ at some generic value, for example $a$, doing this you get $R(T = a) = R_a$

So your function is reduced to $V(i) = i \cdot R_a$.

Now you can easily derive: $$\dfrac{dV(i)}{di} = R_a$$

Using this way a 2 variable function has been reduced to a 1 variable function. Note that $R_a$ is a constant at a fixed $T$ but depends on value you assign to $T$.

Having an extra variable ($T$) can seem confusing. If you have to derive a function with many variables you should rewrite it and show which variables depend on the variable you are deriving.