Algebra: How do I derive this equation given two other equations?

[dsilvas]

Homework Statement

Derive eq.(5) and eq.(6) from eq.(3) and (4)

Relevant Equations

Unsure how to type the equations correctly in this text box (no formating options). They are below.

This image shows the equations.
I managed to almost get equation 5, but my partial derivative is not squared but instead multiplied by mu, and also I don't have a factor of 1/2.
Here is an image of the work I have. I'm sorry for any sloppiness. I tried to be as concise as possible when writing it down but it should be pretty straightforward. Essentially the only way my final equation could be equal to eq.(5) is if mu is equal 1/2 of its partial derivative with respect to r.

If it helps, here is the link to the paper that these equations are coming from. It is page 2.
https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/98RS01523
I didn't write it out explicitly, so to elaborate, r prime in my work is equal to the full derivative of r with respect to theta. I was able to derive eq. (6) from eq. (4) and that has helped me get closer to deriving eq. (5) but I'm not quite there and I'm stuck.

 

[mitochan]

Hi.
In the last line in your notebook substitute
$$\mu \frac{d\mu}{dr}=\frac{1}{2}\frac{d\mu^2}{d\mu}\frac{d\mu}{dr}=\frac{1}{2}\frac{d\mu^2}{dr}$$

 

[dsilvas]

Hi.
In the last line in your notebook substitute
$$\mu \frac{d\mu}{dr}=\frac{1}{2}\frac{d\mu}{dr}$$

Oh gosh. I think I realize my mistake now, thank you.
I assumed in eq 5 that it was $${(\frac{d\mu}{dr})}^2$$ when in reality it was $$\frac{d(\mu^2)}{dr}$$
I just can't read. Thank you. I didn't realize it was that simple.

I guess this question can be closed now!