⛰️ Directional Derivative Calculator

Directional Derivative Calculator

The **Directional Derivative Calculator** is your indispensable tool for multivariable calculus. Unlike standard partial derivatives, which only measure the rate of change parallel to the axes, the directional derivative, $D_{\vec{u}}f(P)$, measures the instantaneous rate of change of a function $f$ at a point $P$ in any arbitrary direction $\vec{v}$. This is crucial for understanding surfaces, temperature changes, or velocity fields in non-axis-aligned directions.

The Formula: Directional Derivative $D_{\vec{u}}f$

The calculation is based on the dot product of the function's **Gradient Vector** and the **Unit Direction Vector**:

$$D_{\vec{u}}f(P) = \nabla f(P) \cdot \vec{u}$$

Where:

• $\mathbf{\nabla f(P)}$ is the gradient vector of $f$, evaluated at the point $P$. For $f(x, y)$, $\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle$.
• $\mathbf{\vec{u}}$ is the unit vector in the direction of the vector $\vec{v}$. It is calculated as $\vec{u} = \frac{\vec{v}}{\|\vec{v}\|}$.

Step-by-Step Calculation Explained

Our calculator follows three fundamental steps to provide the directional derivative:

Step 1: Calculate the Gradient Vector $\nabla f$

The gradient is the vector of all the first partial derivatives of the function. For a function $f(x, y)$, we compute $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$. The gradient always points in the direction of the **maximum rate of increase** of the function.

Step 2: Evaluate the Gradient at Point $P$

Once the symbolic gradient $\nabla f$ is found, we substitute the coordinates of the given point $P(x_0, y_0)$ into the partial derivative expressions to get a numerical vector $\nabla f(P)$.

Step 3: Find the Unit Direction Vector $\vec{u}$

The direction vector $\vec{v}$ must be normalized to a unit vector $\vec{u}$ (a vector with a magnitude of 1). If $\vec{v} = \langle a, b \rangle$, the magnitude is $\|\vec{v}\| = \sqrt{a^2 + b^2}$, and $\vec{u} = \frac{1}{\|\vec{v}\|} \langle a, b \rangle$.

Step 4: Compute the Dot Product

Finally, we multiply the gradient vector $\nabla f(P)$ and the unit vector $\vec{u}$ using the dot product to find the scalar value of the directional derivative $D_{\vec{u}}f(P)$. This value represents the slope of the surface $z=f(x, y)$ in the direction $\vec{u}$ at the point $P$.

Master multivariable calculus concepts like level curves, tangent planes, and gradients by using this fast, accurate directional derivative solver today!