A partial derivative measures change along a coordinate axis. A directional derivative measures change in any chosen direction.

Let $\mathbf u$ be a unit vector. The directional derivative of $f$ at $\mathbf a$ in the direction $\mathbf u$ is

$$D_{\mathbf u}f(\mathbf a)

\lim_{h\to0} \frac{f(\mathbf a+h\mathbf u)-f(\mathbf a)}{h},$$

when this limit exists.

Using the gradient

If $f$ is differentiable, then

$$D_{\mathbf u}f(\mathbf a)

\nabla f(\mathbf a)\cdot\mathbf u.$$

Thus a directional derivative is the component of the gradient along the chosen direction.

For example, if

$$\nabla f(\mathbf a)=(3,4)$$

and

$$\mathbf u=\left(\frac35,\frac45\right),$$

then

$$D_{\mathbf u}f(\mathbf a) =(3,4)\cdot\left(\frac35,\frac45\right)=5.$$

Fastest increase and decrease

Because

$$\nabla f\cdot\mathbf u =|\nabla f|\cos\theta,$$

the directional derivative is largest when $\mathbf u$ points in the same direction as the gradient. Its maximum value is

$$|\nabla f|.$$

The minimum occurs in the opposite direction and equals $-|\nabla f|$.

A direction perpendicular to the gradient gives directional derivative zero, which explains why tangent directions to level sets produce no first-order change in the function.