Section 14.6: Directional Derivatives & Gradient Vector

For functions of three variables we can define directional derivatives in a similar manner. Again \(D_{\mathbf{u}}f(x, y, z)\) can be interpreted as the rate of change of the function in the direction of a unit vector u.

Definition The directional derivative of f at \((x_0, y_0, z_0)\) in the direction of a unit vector \(\mathbf{u} = \langle a, b, c \rangle\) is

\(D_{\mathbf{u}}f(x_0, y_0, z_0) = \lim_{h\to0} \frac{f(x_0 + ha, y_0 + hb, z_0 + hc) - f(x_0, y_0, z_0)}{h}\)

if this limit exists.

Gradients

For a function f of three variables, the gradient vector, denoted by \(\nabla f\) or grad f, is

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

Then, just as with functions of two variables, the formula for the directional derivative can be rewritten as

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

EXAMPLE If \(f(x, y, z) = x \sin(yz)\), (a) find the gradient of f and (b) find the directional derivative of f at \((1, 3, 0)\) in the direction of \(\mathbf{v} = \mathbf{i} + 2\mathbf{j} - \mathbf{k}\).