1. The problem statement, all variables and given/known data Let f: R^n -> R be defined as follows: f(x) = x*L(x) where * denotes the standard inner product and L: R^n -> R^n is a linear function. I'm trying to find the directional derivative f'(x;u). 2. Relevant equations I know that f'(x;u) (the directional derivative of f(x) in the direction of the unit vector u) is equal to gradient(f(x)) * u where * denotes inner product. 3. The attempt at a solution In this case gradient(f(x)) = gradient(x*L(x)) = gradient(x)*L(x) + L(x)*gradient(x) (Not sure if this step is correct). Then because L: R^n -> R^n is a linear map the differential of L is L again, no? So we get: gradient(f(x)) = gradient(x)*L(x) + L(x)*gradient(x) = 2L(x)*gradient(x). Conclusion: the directional derivative in the direction of u is then (2L(x)*gradient(x))*u where u is a unit vector. Is this correct? I'm a little bit confused about the part of L being a linear map, is it correct to state that the gradient is again L itself?