Product Rule of Gradient

 

Let f : ℝⁿ → ℝ and g : ℝⁿ → ℝ be scalar-valued differentiable function of several (n) variables. Then f and g satisfy the product rule (the Leibniz rule):

 

∇[fg] = f∇g + g∇f

or

∇[f(x)g(x)] = f(x)∇g(x) + g(x)∇f(x)

for all x ∈ ℝⁿ.

 

Proof

Given then clearly we have