Partial Derivative over an element 𝑥_𝑖 in input vector 𝐱 = (𝑥₁, …, 𝑥_𝑛)

• $\frac{𝜕f(\text{x})}{𝜕x_i}=li{m}_{h\to 0}\left[\frac{f(x_1,\dots ,x_i+h,\dots ,x_n)-f(x_1,\dots ,x_i,\dots ,x_n)}{(x_i+h)-(x_i)}\right]$

Total Derivative over all variables 𝑥_𝑖s in coordinate direction

• $df(\text{x})={\sum }_{i=1}^n\left[\frac{𝛿f(\text{x})}{𝛿x_i}d{x}_i\right]$

Gradient at point 𝐱 = (𝑥₁, …, 𝑥_𝑛)

• ${\nabla }_{\text{x}}f(\text{x})=\left[\begin{array}{c}\frac{𝜕f(\text{x})}{𝜕x_1}\frac{𝜕f(\text{x})}{𝜕x_2}...\frac{𝜕f(\text{x})}{𝜕x_n}\end{array}\right]$

Directional Derivative at point 𝐱 = (𝑥₁, …, 𝑥_𝑛)
with UNIT direction 𝐯 = (𝑣₁, …, 𝑣_𝑛)

Similar to the Multivariable Chain Rule

• ${\nabla }_{\text{x}\cdot \text{v}}f(\text{x})={\nabla }_{\text{x}}f(\text{x})\cdot \text{v}$
• ${\nabla }_{\text{x}\cdot \text{v}}f(\text{x})=\left[\begin{array}{c}\frac{𝜕f(\text{x})}{𝜕x_1}\frac{𝜕f(\text{x})}{𝜕x_2}...\frac{𝜕f(\text{x})}{𝜕x_n}\end{array}\right]\cdot \left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]$
• ${\nabla }_{\text{x}\cdot \text{v}}f(\text{x})=li{m}_{h\to 0}\left[\frac{f(\text{x}+h\text{v})-f(\text{x})}{h}\right]$