Section 14.6: Directional Derivatives & Gradient Vector

Show that the operation of taking the gradient of a function has the given property. Assume that u and v are differentiable functions of x and y and that a, b are constants. (a) \(\nabla(au + bv) = a\nabla u + b\nabla v\) (b) \(\nabla(uv) = u\nabla v + v\nabla u\) (c) \(\nabla(\frac{u}{v}) = \frac{v\nabla u - u\nabla v}{v^2}\) (d) \(\nabla u^n = nu^{n-1}\nabla u\)