In this paper, we investigate the commutativity of a prime Banach algebra 𝔄 which admits a nonzero continuous linear generalized derivation 𝔉 : 𝔄 → 𝔄 associated with continuous linear derivation δ such that either 𝔉 ([ 𝔵 m , 𝔶 n ]) − [ 𝔉 (𝔵 m) , 𝔶 n ] ∈ 𝒵 (𝔄) or 𝔉 ([ 𝔵 m , 𝔶 n ]) + [ 𝔉 (𝔵 m) , 𝔶 n ] ∈ 𝒵 (𝔄) for intergers m = m (𝔵 , 𝔶) > 1 and n = n (𝔵 , 𝔶) > 1 and sufficiently many 𝔵 , 𝔶 ∈ 𝔄. Further, similar results are also obtained for unital prime Banach algebra 𝔄 which admits a nonzero continuous linear generalized derivations 𝔉 satisfying either 𝔉 (𝔵 𝔶) n − 𝔉 (𝔵 n) 𝔶 n ∈ 𝒵 (𝔄) or 𝔉 (𝔵 𝔶) n − 𝔶 n 𝔉 (𝔵 n) ∈ 𝒵 (𝔄) for an integer n = n (𝔵 , 𝔶) > 1 and sufficiently many 𝔵 , 𝔶. [ABSTRACT FROM AUTHOR]