Section 14.6: Directional Derivatives & Gradient Vector

Definition If f is a function of two variables x and y, then the gradient of f is the vector function \(\nabla f\) defined by

\[ \nabla f(x, y) = \langle f_x(x, y), f_y(x, y) \rangle = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} \]

EXAMPLE If \(f(x, y) = \sin x + e^{xy}\), then

\(\nabla f(x, y) = \langle f_x, f_y \rangle = \langle \cos x + ye^{xy}, xe^{xy} \rangle\)

and

\(\nabla f(0, 1) = \langle 2, 0 \rangle\)

With this notation for the gradient vector, we can rewrite the equation for the directional derivative of a differentiable function as

\[ D_{\mathbf{u}}f(x, y) = \nabla f(x, y) \cdot \mathbf{u} \]

This expresses the directional derivative in the direction of a unit vector u as the scalar projection of the gradient vector onto u.

EXAMPLE Find the directional derivative of the function \(f(x, y) = x^2y^3 - 4y\) at the point \((2, -1)\) in the direction of the vector \(\mathbf{v} = 2\mathbf{i} + 5\mathbf{j}\).