This is the 560th most frequent Russian word.
большей
The Russian word "большей" means "greater" or "larger" (feminine genitive or instrumental singular form of "большой").
In this sentence, "большей" is the feminine, instrumental case form of "больший", used to express a comparative degree, referring to the extent of dependence.
В большей мере успех зависит от опыта.
Success depends on experience to a greater extent.
Here, "большую" is the feminine, accusative case form of "большой" and means "great" or "big", modifying "love" to describe its magnitude.
Она верила в большую любовь.
She believed in great love.
Here, "большим" is a declension of "большой", meaning "great" or "big" in the instrumental case, modifying "achievement" to show comparison or the utmost degree.