# Directional derivatives

For a function of one variables $f(x)$, $f'(x)>1$ implies that the rate of increase of $f(x)$ at $x$ is more than 1. How can we define the analogous notion for a function of two variables $f(x,y)$, i.e. whenever $f(x,y)$ increases, the rate of increase is more than 1?

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If $v$ is a unit vector, the directional derivative of the function $f$ is $$v \cdot \nabla f = |\nabla f| \cos \theta,$$ where $\theta$ is the angle between $v$ and $\nabla f.$ –  Will Jagy Dec 30 '11 at 4:27