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Section 14.6: Directional Derivatives & Gradient Vector

Exercise 40

  1. If \(\mathbf{u} = \langle a, b \rangle\) is a unit vector and f has continuous second partial derivatives, show that \(D^2_{\mathbf{u}}f = f_{xx}a^2 + 2f_{xy}ab + f_{yy}b^2\)
  2. Find the second directional derivative of \(f(x, y) = xe^{2y}\) in the direction of \(\mathbf{v} = \langle 4, 6 \rangle\).