Let A be a second order tensor and let f ( A ) = det ( A ) . Then, from the definition of the derivative of a scalar valued function of a tensor, we have ∂ ∂ f ∂ ∂ A : T = d d α α det ( A + α α T ) | α α = 0 = d d α α det [ α α A ( 1 α α I + A − − 1 ⋅ ⋅ T ) ] | α α = 0 = d d α α [ α α 3 det ( A ) det ( 1 α α I + A − − 1 ⋅ ⋅ T ) ] | α α = 0 .