Looks to me like you have copied incorrectly- you should not have that on the right side of the equation defining the function .
If then we use the chain rule:
Letting , .
Then and
By the chain rule then
Good morning all, I am following a book which covers aspects of astronomy/cosmology, I'd like to know how the part in the middle is worked out. I have the start and end of the derivation but its quite scant on how it proceeds from point to point, the book simply says using the chain rule "this" becomes "this", basically.
∂ρ/∂z
Where the solution for is:
The result is:
This is simply for myself and trying to understand how the calculations are derived.
Thanks in advance
Rod.
It could probably be my copying, forgive an old man I am not well adjusted to using the symbol format on the forum. When I retired so it seems did my eyesight and finger control.
I'll try and be more accurate:
∂ρ/∂z
Where the solution for is:
Then it jumps to:
∂ρ/∂z =
Writing we have [tex]\rho(u)= \rho_ce^{u}[/itex] so the
Do you understand that? It is simply asserting that the derivative of the exponential is the exponential and is a defining property of the exponential.
Also . That is just using the fact that the derivative of z squared is 2z.
Then, by the chain rule
but that first parentheses is so that