Chain or product rule for heat diffusion equation

Question

A portion of the heat diffusion equation for a 1-D solid is given as:

$$\frac{1}{r} \frac{\partial}{\partial r} \left(r \; k \frac{\partial T}{\partial r} \right)$$

Apparently this can be expanded into the following form using the Chain rule:

$$\frac{\partial}{\partial r} \left(k \frac{\partial T}{\partial r} \right) + \frac{1}{r} \left(k \frac{\partial T}{\partial r} \right)$$

However, the only way I can get this form is using the Product rule, not the Chain rule.

Is there a way to use the Chain rule or is the Product rule actually the method to use?

Answer 1

This is definitely the product rule. Note that $$\frac{1}{r} \frac{\partial}{\partial r} \left(r \; k \frac{\partial T}{\partial r} \right)={1\over r}\left(\frac{\partial}{\partial r}(r)\cdot\left(k \frac{\partial T}{\partial r}\right)+\frac{\partial}{\partial r}\left(k \frac{\partial T}{\partial r}\right)(r)\right)=\frac{\partial}{\partial r} \left(k \frac{\partial T}{\partial r} \right) + \frac{1}{r} \left(k \frac{\partial T}{\partial r} \right)$$