Gradient And Directional Derivative Tool — Theory
1. What is the gradient?
The gradient of a multivariable function is the vector made from its partial derivatives.
For a function of two variables,
\[
\nabla f(x,y)
=
\left\langle
f_x,\,
f_y
\right\rangle.
\]
For a function of three variables,
\[
\nabla f(x,y,z)
=
\left\langle
f_x,\,
f_y,\,
f_z
\right\rangle.
\]
2. Meaning of the gradient
The gradient points in the direction where the function increases most rapidly.
Its magnitude gives the maximum rate of increase.
\[
\max D_{\mathbf{u}}f
=
\|\nabla f\|.
\]
3. Unit direction vector
A directional derivative must use a unit vector. If the direction vector is
\(\mathbf{v}\), then the corresponding unit vector is
\[
\mathbf{u}
=
\frac{\mathbf{v}}{\|\mathbf{v}\|}.
\]
For example, if
\[
\mathbf{v}=\langle 3,4\rangle,
\]
then
\[
\|\mathbf{v}\|
=
\sqrt{3^2+4^2}
=
5,
\qquad
\mathbf{u}
=
\left\langle
\frac35,\frac45
\right\rangle.
\]
4. Directional derivative formula
The directional derivative of \(f\) in the unit direction \(\mathbf{u}\) is
\[
D_{\mathbf{u}}f
=
\nabla f\cdot\mathbf{u}.
\]
This dot product measures how much of the gradient points in the chosen direction.
5. Example
Let
\[
f(x,y)=x^2+y^2.
\]
The partial derivatives are
\[
f_x=2x,
\qquad
f_y=2y.
\]
Therefore,
\[
\nabla f=\langle 2x,2y\rangle.
\]
At \((1,2)\),
\[
\nabla f(1,2)=\langle 2,4\rangle.
\]
In the direction \(\mathbf{v}=\langle 3,4\rangle\),
\[
\mathbf{u}
=
\left\langle
\frac35,\frac45
\right\rangle.
\]
So the directional derivative is
\[
\begin{aligned}
D_{\mathbf{u}}f
&=
\langle 2,4\rangle
\cdot
\left\langle
\frac35,\frac45
\right\rangle\\
&=
2\cdot\frac35
+
4\cdot\frac45\\
&=
4.4.
\end{aligned}
\]
6. Maximum increase and decrease
The maximum increase occurs in the direction of the gradient:
\[
\mathbf{u}_{\max}
=
\frac{\nabla f}{\|\nabla f\|}.
\]
The maximum decrease occurs in the opposite direction:
\[
\mathbf{u}_{\min}
=
-\frac{\nabla f}{\|\nabla f\|}.
\]
7. When the gradient is zero
If
\[
\nabla f=\mathbf{0},
\]
then the function has no preferred direction of greatest increase at that point.
The maximum directional derivative is \(0\).
8. Angle interpretation
The dot product can also be written as
\[
D_{\mathbf{u}}f
=
\|\nabla f\|\cos\theta,
\]
where \(\theta\) is the angle between \(\nabla f\) and \(\mathbf{u}\).
The directional derivative is largest when \(\theta=0\).
9. Vector field interpretation
The vector field shown by the calculator places a gradient vector at many points.
These arrows show the local direction of steepest increase.
\[
(x,y)
\mapsto
\nabla f(x,y).
\]
The selected direction arrow shows the direction used in the directional derivative.
11. Common mistakes
- Forgetting to normalize: standard directional derivatives use a unit vector.
- Using the raw direction vector: \(\nabla f\cdot\mathbf{v}\) is not the standard directional derivative unless \(\mathbf{v}\) is unit length.
- Confusing gradient with direction: the gradient is determined by the function, while \(\mathbf{u}\) is the chosen direction.
- Ignoring the point: the gradient must be evaluated at the selected point.
- Misreading negative values: a negative directional derivative means the function decreases in that direction.
