Directional Derivative

Theorem

• Suppose that $f:{\mathbb{R}}^n\to \mathbb{R}$
• We can define a derivative $f$ at $x$ in direction $v$ to be:
• $Df(x)v=\frac{d}{dt}f(x+tv){|}_{t=0}={\lim }_{t\to 0}\frac{f(x+y)-f(x)}{t}=\nabla f\cdot v$

Directional Derivative with Gradient

$Df(x)v=\nabla f\cdot v$

Finding Directional Derivative Process

Given:

• A function $f$
• A origin point $x_0$
• A direction vector $v$

1. Find $\nabla f$
2. Evaluate $\nabla f(x_0)$
3. Find the normal vector $\frac{v}{||v||}$
4. Evaluate $\nabla f(x_0)\cdot \frac{v}{||v||}$, this is your directional derivative