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

Also known as 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.

Written by

Sahithyan K.

October 19, 2026 1 min read

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