Directional Derivative

Rate of change of a multivariable function in the direction of the unit vector $\boldsymbol{u}=(a,b)$. $$

Directional derivative of $f$ in the direction $\boldsymbol{u}$ is:

$$D_{\boldsymbol{u}} f(x_0,y_0) = \lim_{h\to 0} \frac{\Delta z}{h} = \lim_{h\to 0} \frac{f(x_0+ha, y_0+hb) - f(x_0,y_0)}{h}$$

If $f_x$ and $f_y$ are continuous at $(x,y)$, then $f$ has a directional derivative in any direction $\boldsymbol{u}=(a,b)$.

$$D_{\boldsymbol{u}} f(x,y) = af_x(x,y) + bf_y(x,y)$$

Also the directional derivative can be written as:

$$D_{\boldsymbol{u}} f(x_0,y_0) = \Big\langle f_x(x_0,y_0),f_y(x_0,y_0)\Big\rangle \cdot \boldsymbol{u}$$

Gradient

Denoted by $\nabla f$. Can be extended for functions with more inputs. $$

$$\nabla f(x_0,y_0) = \Big\langle f_x(x_0,y_0),f_y(x_0,y_0) \Big\rangle = \frac{\partial f}{\partial x}\boldsymbol{i} +\frac{\partial f}{\partial y}\boldsymbol{j}$$

The $\nabla$ is the “del operator”. $$

$$\nabla \equiv \frac{\partial }{\partial x}\boldsymbol{i} +\frac{\partial }{\partial y}\boldsymbol{j}$$

Critical point

Aka stationary point. A point where the gradient is zero or where one of the partial derivatives is undefined.

Maxmimum of Directional Derivative

Maximum value of the directional derivative D_\boldsymbol{u} f is $\lvert \nabla f \rvert$ and occurs when the gradient vector and $\boldsymbol{u}$ has the same direction.