Not sure how it would work out, but I'd be tempted to differentiate using the product rule, the various terms being:
, , .
Then I'd see what values of make it so that what you get has a repeated root, i.e. only one value of the resulting equation equal to zero. It will probably be a quadratic in .
Splitting into partial fractions is all very well, but I thought this was only usually done when integrating. In this case you're differentiating.