> [!definition] > > Let $E, F$ be [[Topological Vector Space|topological vector spaces]], $U \subset E$ and $V \subset F$ be open sets, $f: U \to V$ be any [[Function|function]]. The **directional derivative** of $f$ at $v$ is > $ > D_vf(x_0) = \lim_{t \to 0}\frac{F(x_0 + tv) - F(x_0)}{t} > $ > if the [[Limit|limit]] exists. > [!definition] > > $ > D_\hat{u}f = \hat{u} \cdot \text{grad}\ f > $ > Let $f(x_1, \cdots, x_n): \real^n \to \real$ be a [[Derivative|differentiable]] function of $n$ variables. The **directional** [[Derivative|derivative]] of $f$ at a certain point in a direction, specified by a unit [[Vector|vector]], is the change in the function's value with respect to change in that direction, calculated as the [[Inner Product|dot product]] between the unit vector and its [[Gradient|gradient]]. > [!theorem] > > By the property of the [[Inner Product|dot product]], the directional derivative is maximised when the direction $\hat{u}$ is parallel to the gradient.