Help with deriving simple heat equation [closed]

Question

$$j^{q}=\frac{1}{2} n v[\varepsilon(T[x-v \tau])-\varepsilon(T[x+v \tau])]$$

To this:

$$j^{q}=n v^{2} \tau \frac{d \varepsilon}{d T}\left(-\frac{d T}{d x}\right)$$

At first I was thinking of using the fundamental theorem of calculus but I can't seem to do it. Any words of advice would be appreciated.

Answer 1

Use:

$$\frac{d\epsilon}{dT}=\frac{\epsilon(T+dT)-\epsilon(T-dT)}{2dT}$$

and:

$$dT=\frac{dT}{dx}dx=\frac{dT}{dx}v\tau$$