14.2 | Computing Partial Derivatives Algebraically


Consider the surface defined as the level set \(x^3+y^3+z^3+4xyz = 0\), and \(P(1,-1,2)\) a point on this surface. Assume that close to \(P\) the surface determines an implicit function \(z = h(x,y)\).

  1. Determine the gradient of \(h\) at \(P\).
  2. Determine a direction \(P\) (a direction with respect to \(x\) and \(y\)) such that the rate of change of \(h\) is zero.

  1. $$
    \begin{aligned}
    \frac{d}{dx}[x^3 +y^3 + z^3 + 4xyz] & = \frac{d}{dx}[0] \\
    \frac{d}{dx}[x^3] +\frac{d}{dx}[y^3] + \frac{d}{dx}[z^3] + \frac{d}{dx}[4xyz] & = 0 \\
    3x^2 + 3z^2\frac{dz}{dx}+4xy\frac{dz}{dx} & = 0 \\
    \frac{dz}{dx}(3z^2+4xy) & = -3x^2-4yz \\
    \frac{dz}{dx} & = \frac{-3x^2-4yz}{3z^2+4xy}
    \end{aligned}
    $$
    or
    $$
    \begin{aligned}
    \frac{dz}{dx} & = -\frac{\frac{dF}{dx}}{\frac{dF}{dz}} \\
    & = -\frac{3x^2+4yz}{3z^2+4xy}
    \end{aligned}
    $$
    where \(F(x,y,z) = x^3 +y^3 + z^3 + 4xyz\). So,
    $$
    h_x = -\frac{3x^2+4yz}{3z^2+4xy}.
    $$
    Similarly,
    $$
    h_y = -\frac{3y^2 + 4xz}{3z^2+4xy}.
    $$
    So, \(\nabla h = \frac{1}{3z^2+4xy} \langle 3x^2+4yz, 3y^2+4xz \rangle\) and \(\nabla h(1,-1,2) = \langle 5/8,-11/5 \rangle\).

E 14.2 Exercises