Why there is a sudden change from positive to negative after changing $dt$ to $d\sigma$?

Question

I wonder why there is a negative sign after changing $dt$ to $d\sigma$ on the third line.

Answer 1

Making sku's comment more explicit, consider $g$ as the function composition $(g \circ y)(c)$, where $y$ is everything inside of $g$ and $c = t-\sigma$. By the chain rule, $$ {\mathrm{d}g \over \mathrm{d}t} = {\mathrm{d}g \over \mathrm{d}y} {\mathrm{d}y \over \mathrm{d}c} {\partial c \over \partial t} = {\mathrm{d}g \over \mathrm{d}y} {\mathrm{d}y \over \mathrm{d}c} $$ and $$ {\mathrm{d}g \over \mathrm{d}\sigma} = {\mathrm{d}g \over \mathrm{d}y} {\mathrm{d}y \over \mathrm{d}c} {\partial c \over \partial \sigma} = -{\mathrm{d}g \over \mathrm{d}y} {\mathrm{d}y \over \mathrm{d}c} $$ therefore, $$ {\mathrm{d}g \over \mathrm{d}t} = -{\mathrm{d}g \over \mathrm{d}\sigma} $$