I have a function:
$$G=\frac{(\frac{4\pi^2}{T^2}+\frac{1}{\tau^2})\cdot r^2 \cdot d\cdot S}{4M\cdot L}$$
and I need to do error propagation on this function:
$$\delta G=\sqrt{(\frac{\partial G}{\partial \tau}\delta \tau)^2+(\frac{\partial G}{\partial r}\delta r)^2+(\frac{\partial G}{\partial d}\delta d)^2+(\frac{\partial G}{\partial L}\delta L)^2+(\frac{\partial G}{\partial S}\delta S)^2+(\frac{\partial G}{\partial M}\delta M)^2+(\frac{\partial G}{\partial T}\delta T)^2}$$
If by looking at this you already have a headache...Yeah, I'm already dying. So I know how to do this for simpler functions like
$$f(m,v)=\frac{1}{2}mv^2$$ $$\delta f=\sqrt{(\frac{\partial f}{\partial m}\delta m)^2(\frac{\partial f}{\partial v}\delta v)^2}$$
Where $\frac{\partial f}{\partial m}=\frac{1}{2}v^2$ and $\frac{\partial f}{\partial v}=mv$...Or cases for addition.
But the giant thing I have here where there is adition, multiplication, and powers, i'm just lost. How would the idividual term look like?
Logarithmic differentiation is your friend $$G=\frac{\left(\frac{4 \pi ^2}{T^2}+\frac{1}{\tau^2}\right)\, r^2 \, d\, S}{4M\, L}=\frac{\left(\frac{4 \pi ^2 \tau ^2}{T^2}+1\right)\, d\, S}{4M\, L}$$ $$\log(G)=\log\left(\frac{4 \pi ^2 \tau ^2}{T^2}+1\right)+\log(d)+\log(S)-\log(M)-\log(L)-\log(4)$$ Differentiate to obtain, for any variable $X$,$$\frac1 G\frac{\partial G}{\partial X}=\text{something}$$ and, when done,use $$\frac{\partial G}{\partial X}=G \times\text{something}$$