Section 4.6 - Directional Derivatives and the Gradient Vector

Section 4.6 - Directional Derivatives and the Gradient Vector

Partial Derivatives

Partial derivatives measure the rate of change of a multivariable function with respect to one variable, keeping the others constant. For a function $f(x, y)$, the partial derivatives are defined as:

• Partial derivative with respect to x: $f_x(x_0, y_0) = \lim_{h \to 0} \frac{f(x_0 + h, y_0) - f(x_0, y_0)}{h}$

• Partial derivative with respect to y: $f_y(x_0, y_0) = \lim_{h \to 0} \frac{f(x_0, y_0 + h) - f(x_0, y_0)}{h}$

These derivatives represent the rates of change in the $x$ and $y$ directions, respectively.

Directional Derivative

The directional derivative of $f$ at a point $(x_0, y_0)$ in the direction of a unit vector $\vec{u} = (a, b)$ is the rate at which $f$ changes as we move from $(x_0, y_0)$ in the direction of $\vec{u}$.

• Definition: $D_{\vec{u}}f(x_0, y_0) = \lim_{h \to 0} \frac{f(x_0 + ah, y_0 + bh) - f(x_0, y_0)}{h}$

• If $\vec{u}$ makes an angle $\theta$ with the positive $x$-axis, then $\vec{u} = (\cos\theta, \sin\theta)$ and: $D_{\vec{u}}f(x, y) = f_x(x, y)\cos\theta + f_y(x, y)\sin\theta$

Special Cases

• If $\vec{u} = (1, 0)$, then $D_{\vec{u}}f = f_x$

• If $\vec{u} = (0, 1)$, then $D_{\vec{u}}f = f_y$

Thus, partial derivatives are special cases of the directional derivative.

Theorem: Existence of Directional Derivative

• If $f$ is differentiable at $(x, y)$, then $f$ has a directional derivative in the direction of any unit vector $\vec{u} = (a, b)$, given by: $D_{\vec{u}}f(x, y) = f_x(x, y)a + f_y(x, y)b$

Examples

• Example 1: Find the directional derivative $D_{\vec{u}}f(x, y)$ if $f(x, y) = 3x^2 + 4y^2$, and $\vec{u}$ is the unit vector given by angle $\theta = \pi/6$. Solution: $f_x(x, y) = 6x$, $f_y(x, y) = 8y$, $\vec{u} = (\cos(\pi/6), \sin(\pi/6))$.

• Example 2: If $f(x, y) = \sin x + e^{y}$, then $\nabla f(x, y)$ and $D_{\vec{u}}f$ in direction $(0, 1)$ are $f_x = \cos x$, $f_y = e^{y}$, $D_{\vec{u}}f = e^{y}$.

• Example 3: Find the directional derivative of $f(x, y) = x^2 y^3 - 4y$ at the point $(2, -1)$ in the direction of the vector $\vec{v} = 2\vec{i} + 5\vec{j}$. Solution: Normalize $\vec{v}$ to get the unit vector, then apply the formula above.

Gradient Vector

The gradient vector of $f$ at $(x, y)$ is defined as:

• $\nabla f(x, y) = (f_x(x, y), f_y(x, y))$

The directional derivative can be rewritten as a dot product:

• $D_{\vec{u}}f(x, y) = \nabla f(x, y) \cdot \vec{u}$

Functions of Three Variables

For functions of three variables $f(x, y, z)$, the directional derivative in the direction of a unit vector $\vec{a} = (a, b, c)$ is:

• $D_{\vec{a}}f(x_0, y_0, z_0) = \lim_{h \to 0} \frac{f(x_0 + ah, y_0 + bh, z_0 + ch) - f(x_0, y_0, z_0)}{h}$

• Or, $D_{\vec{a}}f(x_0, y_0, z_0) = \nabla f(x_0, y_0, z_0) \cdot \vec{a}$

Example

• Example 4: If $f(x, y, z) = \sin(xz)$, find the gradient of $f$ and the directional derivative of $f$ at $(1, 3, 0)$ in the direction of $\vec{v} = 7\vec{i} + 2\vec{j} - \vec{k}$.

Maximizing the Directional Derivative

For a differentiable function $f$ of two or three variables, the maximum value of the directional derivative at a point occurs when the direction is the same as the gradient vector $\nabla f$.

• Properties of the Gradient:

  • If $\nabla f(x_0, y_0) = 0$, then $D_{\vec{u}}f(x_0, y_0) = 0$ for any unit vector $\vec{u}$.

  • If $\nabla f(x_0, y_0) \neq 0$, then $D_{\vec{u}}f(x_0, y_0)$ is maximized when $\vec{u}$ points in the same direction as $\nabla f(x_0, y_0)$. The maximum value is $|\nabla f(x_0, y_0)|$.

  • $D_{\vec{u}}f(x_0, y_0)$ is minimized when $\vec{u}$ points in the opposite direction. The minimum value is $-|\nabla f(x_0, y_0)|$.

Theorem: Suppose $f$ is a differentiable function of two or three variables. The maximum value of the directional derivative $D_{\vec{u}}f$ is $|\nabla f|$ and it occurs when $\vec{u}$ has the same direction as the gradient vector $\nabla f$.

Significance of the Gradient Vector

The gradient vector $\nabla f(x_0, y_0)$ at a point $(x_0, y_0)$ in the domain of $f$:

• Gives the direction of fastest increase of $f$.

• Is orthogonal (perpendicular) to the level surface $S$ of $f$ through $(x_0, y_0)$.

Moving away from $P$ on the level surface $S$, the value of $f$ does not change; moving in the direction of $\nabla f$ gives the maximum increase.

Concept	Definition	Formula
Partial Derivative	Rate of change in one variable, others constant	$f_x(x, y), f_y(x, y)$
Directional Derivative	Rate of change in direction of unit vector $\vec{u}$	$D_{\vec{u}}f(x, y) = \nabla f(x, y) \cdot \vec{u}$
Gradient Vector	Vector of partial derivatives	$\nabla f(x, y) = (f_x(x, y), f_y(x, y))$
Maximum Directional Derivative	Occurs in direction of $\nabla f$	$|\nabla f|$

Additional info: The notes include graphical illustrations and examples to clarify the geometric meaning of the gradient and directional derivatives, as well as their applications in multivariable calculus.