Good typesetting practice for long equations

breaking before not after operators and defining names for the subterms

\documentclass{article}
\usepackage{amsmath}

\begin{document}


\begin{align*}
\frac{\partial^2}{\partial t_1^2} f(t_0,t_1)
 &= 
b^{a-1} \cdot \bigl(  
\frac{\partial^2}{\partial t_1^2}a \cdot b \cdot  \log ( b) +
a' \cdot 2 \cdot  \log ( b)+
a' \cdot b \cdot  \frac{2}{b} +
2 \frac{\partial}{\partial t_1} a \bigr) \\
 &\quad+
 b^{a-2}\cdot
 \bigl( \frac{\partial}{\partial t_1}a \cdot b \cdot \log ( b) + (a -2) \bigr) \cdot 
\bigl( a' \cdot b \cdot  \log ( b) + 2a\bigr)\\
  & = 
b^{a-1} \cdot \Bigl( 
 \frac{\partial^2}{\partial t_1^2}a \cdot  b \cdot  \log ( b) +
2 \cdot a'  \cdot  \bigl( 2 + \log ( b) \bigr) \Bigr)\\
&\quad +
b^{a-2} \cdot \bigl(a' \cdot 
c \cdot \log (c) +  
\bigl(a -2) \bigr) \cdot
 \bigl(a' \cdot  b \cdot  \log ( b) +2a)\bigr)\bigr)\\
  &< 0
\end{align*}
where:\\
$a=\alpha( w-t_0+t_1 )$\\
$a'=\alpha'(w-t_0+t_1)$\\
$b=\delta+2t_0+2t_1$\\
$c=\delta + 2t_0+2t_1$
\end{document}

I'd try to make the equation smaller by grouping parts:

• Don't use \cdot where it's not necessary. I use it only for scalar products of vectors and for numbers, but not for symbolic factors or before parentheses.
• Derivatives are often written as \partial_{t_1} instead of \frac{\partial}{\partial t_1}. This can save some space.
• Introducing substitutions can be helpful. In your code (\delta+2t_0+2t_1) appears quite often and it could be replaced by a new symbol which will be defined before or after the equation
• Align the equation at least on all equal signs: &=
• Other line breaks may be before + signs to "group" summands (this shows that the equation consists of similar parts that are added together)

Actually, I would like to start answering with a question: Is it very informative to display an equation that long?

I would try to identify parts in your equation, and write something like

\[a (A + B + C) < 0\]
where
\[a = ... \]
and
\begin{align} 
A &= ... \\
B &= ... \\
C &= ...
\end{align}

this makes it much easier to read it, and you can maybe give also an exlanation to every term.