Prove the Product Rule for Gradients in Theorem 1. THEOREM 1 Properties of the Gradient If f(x,

Prove the Product Rule for Gradients in Theorem 1.

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THEOREM 1 Properties of the Gradient If f(x, y, z) and g(x, y, z) are differentiable and c is a constant, then (i) V(f + g)= Vf + Vg (ii) V(cf) = cVf (iii) Product Rule for Gradients: V(fg) = fVg+gVf (iv) Chain Rule for Gradients: If F(t) is a differentiable function of one variable, then V(F(f(x, y, z))) = F'(f(x, y, z))Vf