If the formula for sustainable growth rate is
g = p(1-d)(1+L)/(T-p(1-d)(1+L)
where p = net profit margin, d= dividend payout ratio, T = total assets / total sales, L = debt/equity ratio.

How can I show that it can be expected that the growth rate will increase whenever profit margin increases? I think it is related to finding the derivative or something, but not sure.

The derivative would be one way to do it, but there is an easier way. Note that if 'p' gets bigger the numerator gets bigger, due to the 'p' term there. And also the denominator gets smaller, due to the '-p' term there. Both of these therefore contribute to 'g' increasing.

But the bottom is T - p(1-d)(1+L), so if the second part is greater than T, the result would be negative meaing growth rate is not increasing. How could this be done with the derivative?

Yes, if p > T/(1-d)(1+L) you would have a negative number in the denominator, but for ever larger values of p it becomes less negative, hence it's always increasing. But it must first go through a point where p = T/((1-3d)(1+L)), and at that point growth is infinite. Not knowing what all the variables are meant to represent I can't tell you whether that's a reasonable value for p or not, but I rather doubt it. If you plot this function you get the graph below - to so this I found it easist to replace the quantity (1-d)(1+L) with a constant A, so you have:



Note the discontinuity at p = T/A.

If you are familiar with derivatives you have:



Note the derivative goes to infinity at p = T/A. For all other values the derivative is positive.