Section 4.6 - Directional Derivatives and the Gradient Vector

Section Objectives

1. Compute directional derivatives and interpret them as slopes.
2. Compute gradient vectors.
3. Use gradient vectors as normal vectors.
4. Use gradient vectors to determine the direction of maximum increase/decrease.

Directional derivatives

As we discussed earlier, the partial derivatives give slopes in the directions of the coordinate axes. For slopes in other directions, we need the directional derivative.

Definition:

Suppose $f$ is a two-variable function defined in a neighborhood of $(x,y)$. The directional derivative of $f$ at $(x,y)$ in the direction of $\vec{u}=\cos \theta \, \hat{\imath} + \sin \theta \, \hat{\jmath}$ is given by

provided the limit exists.

Notice that the directional derivative reduces to our usual partial derivatives if $\theta = 0$ or $\theta = \pi/2$.

Theorem:

Suppose $f(x,y)$ is a function such that $f_x$ and $f_y$ exist in a region $R$. At any $(x,y)$ in $R$, the directional derivative of $f$ in the direction of $\vec{u}=\cos \theta \, \hat{\imath} + \sin \theta \, \hat{\jmath}$ is given by

Examples

• Let $f(x,y)=x^3-xy+3y^2$. Find the directional derivative of $f$ at $(-1,2)$ in the direction of $\vec{u}= \cos(\pi/3) \, \hat{\imath} + \sin(\pi/3) \, \hat{\jmath}$.

• Find the directional derivative of $f(x,y)=x^2 \sin y$ at $(1,\pi/2)$ in the direction of $\vec{v}=\langle 3, -4 \rangle$.

The gradient vector

Let $f(x,y)$ be a function whose partial derivatives exist. The gradient of $f$ is defined by

Notice that with this definition, the theorem above shows that the directional derivative can be computed by the dot product

or more generally by

Examples

• Let $g(x,y)=\sin3x \, \cos 5y$. Find $\nabla g(x,y)$.

• Let $f(x,y)=3x^2-2y^2$. Compute the directional derivative of $f$ at $(-3/4,0)$ in the direction from $P\,(-3/4,0)$ to $Q\,(0,1)$.

Properties of the gradient vector

Suppose $f$ is differentiable at $(x,y)$.

1. If $\nabla f(x,y)=\vec{0}$, then $D_{\vec{u}} f(x,y) = 0$ for any unit vector $\vec{u}$.
2. If $\nabla f(x,y) \ne \vec{0}$, the $D_{\vec{u}} f(x,y)$ is maximized when $\vec{u}$ has the same direction as $\nabla f(x,y)$. The maximum value of $D_{\vec{u}} f(x,y)$ is $\|\nabla f(x,y)\|$.
3. If $\nabla f(x,y) \ne \vec{0}$, the $D_{\vec{u}} f(x,y)$ is minimized when $\vec{u}$ has the same direction as $-\nabla f(x,y)$. The minimum value of $D_{\vec{u}} f(x,y)$ is $-\|\nabla f(x,y)\|$.

Examples

• Let $f(x,y)=3x^2-4xy+2y^2$. Find the direction of maximum increase in $f$ at the point $(-2,3)$. What is the slope in that direction?

• The temperature at the point $(x,y)$ on a metal plate in the $xy$-plane is given by $\displaystyle T(x,y)=\frac{xy}{1+x^2+y^2}$, where $T$ is measured in degrees Celsius. An ant at the point $(1,1)$ wants to walk in the direction in which the temperature decreases most rapidly. Find a unit vector in that direction.

Gradients and level curves

Theorem:

Suppose $f(x,y)$ has continuous first-order partial derivatives in a neighborhood of the point $(x_0,y_0)$. If $\nabla f(x_0,y_0) \ne \vec{0}$, then $\nabla f(x_0,y_0)$ is normal to the level curve through $(x_0,y_0)$.

Examples

• Let $f(x,y)=y-\sin x$. Sketch the level curve $f(x,y)=0$. Also sketch $\nabla f(0,0)$ and $\nabla f(\pi/2,1)$.

• Let $f(x,y)=2-x^2-y^2$. Compute a unit vector normal to the level curve $f(x,y)=1$ and $(0,1)$.

Three-dimensional directional derivatives and gradients

The definitions and theorems described above for two-variable functions can be generalized in rather ways to three-variable functions. In particular...

1. $\nabla f(x,y,z)=f_x(x,y,z) \, \hat{\imath} + f_y(x,y,z) \, \hat{\jmath} + f_z(x,y,z) \, \hat{k}$
2. $\displaystyle D_\vec{v} f(x,y,z) = \frac{1}{\|\vec{v}\|}\left[ \nabla f(x,y,z) \cdot \vec{v} \right]$
3. $\nabla f(x,y,z)$ points in the direction most rapid increase in $f$.
4. $-\nabla f(x,y,z)$ points in the direction most rapid decrease in $f$.
5. $\nabla f(x,y,z)$ is normal to the level surface through $(x,y,z)$.

Examples

• Let $f(x,y,z)=5x^2-2xy+y^2-4yz+z^2+3xz$. Find the directional derivative of $f$ at $(1,-2,3)$ in the direction of $\vec{v} = -\hat{\imath} +2 \hat{\jmath} + 2 \hat{k}$.

• Find a vector normal to the surface $x^2+y^2+z^2=6$ at the point $(1,2,1)$. Then find a set of parametric equations for the normal line.

• Find an equation of the plane tangent to the surface $xyz=6$ at $(2,3,1)$.

• Find an equation of the plane tangent to the graph of $z^2-2x^2-2y^2=12$ at the point $(1,-1,4)$. (For review, describe the graph of the equation.)

• Let $f(x,y)=\sqrt{x^2+3y^2}$. Find an equation of the plane tangent to the graph of $z=f(x,y)$ at the point where $(x,y)=(1,1)$.