Chapter 12.6, Problem 81E

Rules for gradients Use the definition of the gradient (in two or three dimensions), assume that f and g are differentiable functions on ¡² or ¡³, and let c be a constant. Prove the following gradient rules.

 a.    Constants Rule: ▿ (cf) = c▿f

 b.    Sum Rule: ▿ (f + g) = ▿f + ▿g

 c.    Product Rule: ▿ (fg) = (▿f)g + f▿g

 d.    Quotient Rule: $\nabla \left(\frac{f}{g}\right)=\frac{g\nabla f-f\nabla g}{g^2}$

e.    Chain Rule: $\nabla \left(f\circ g\right)=f’\left(g\right)\nabla g$, where f is a function of one variable