Section 3.6 Directional Derivative Gradient

3.6 Directional Derivative Gradient

In previous section, we learn take derivative of a function $z=f(x,y)$ with respect to $x$ or with respect to $y$. We can think about the partial derivative with respect to the direction $<1,0>$ and $<0,1>$. In this section we learn to take derivative at any direction. 

Definition: Directional Derivative

Suppose $z=f(x,y)$ is a function of two variables with a domain of $D$. Let $(x_0,y_0)$ be a point in the domain of $z$ and $\overrightarrow{u}=<a,b>$ be an unit vector. Then the directional derivative of $f$ in the direction of $\overrightarrow{u}$ is given by

$$D_{\overrightarrow{u}}f(x_0,y_0)=\underset{h\to 0}{lim}\frac{f(x_0+ah,y_0+bh)-f(x_0,y_0)}{h}$$

provided the limit exists.

Theorem: Directional Derivative of a Function of Two Variables 

Let $z=f(x,y)$ be a function of two variables $x$ and $y$, and assume that $f_x(x,y)$ and $f_y(x,y)$ exist. Then the directional derivative of $f$ in the direction of unit vector $\overrightarrow{u}=<a,b>$ is given by 

$$D_{\overrightarrow{u}}f(x,y)=f_x(x,y)a+f_y(x,y)b.$$

Proof: chain rule. 

Example 1: Finding the directional derivative of $z=f(x,y)=x^{2}y-x{y}^2+2x+y$ in the direction of $\overrightarrow{u}=<\frac{1}{2},\frac{\sqrt{3}}{2}>$.

Exercise 1: Finding the directional derivative of $z=f(x,y)=2x^2+x^{2}y-y^3$ in the direction of $\overrightarrow{u}=<\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}>$.

Remark: Any unit vector $\overrightarrow{u}=<a,b>$ can be rewritten as $\overrightarrow{u}=<\text{cos}(\theta ),\text{sin}(\theta )>$. The directional derivative can only be defined for an unit vector. 

Example 2: Finding the directional derivative of $z=f(x,y)=x^3-2xy+y^2+2x-y$, in the direction of $\overrightarrow{u}=<3,4>$.

Exercise 2: Finding the directional derivative of $z=f(x,y)=x^2+3x^{2}y-y^2+4x+y$, in the direction of $\overrightarrow{u}=<3,3\sqrt{3}>$.

Remark: The directional derivative formula can be rewritten as $$D_{\overrightarrow{u}=<a.b>}f(x,y)=f_x(x,y)a+f_y(x,y)b$$ $$=<f_x(x,y),f_y(x,y)>\cdot <a,b>.$$

Definition: 

Let $z=f(x,y)$ be a function of $x$ and $y$ such that $f_x$ and $f_y$ exist. The vector $\mathrm{\nabla }f(x,y)$ is called the gradient of $f$ and is defined as 

$$\mathrm{\nabla }f(x,y)=<f_x(x,y),f_y(x,y)>$$ 

The vector $\mathrm{\nabla }f(x,y)$ is also written as \textquotedblleft grad $f$ .\textquotedblright{}

Example3: Find the gradient $\mathrm{\nabla }f(x,y)$ where $f(x,y)=\text{sin}(2x)e^y$.

Exercise 3: Find the gradient $\mathrm{\nabla }f(x,y)$ where $f(x,y)=\text{ln}(y)\text{cos}(2x)$.

Theorem: Properties of the Gradient

Suppose the function $z=f(x,y)$ is differentiable at $(x_0,y_0)$.

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

ii. If $\mathrm{\nabla }f(x_0,y_0)\ne 0$, then $D_{\overrightarrow{u}}f(x_0,y_0)$ is maximized when $\overrightarrow{u}$ points in the same direction as $\mathrm{\nabla }f(x_0,y_0)$. The maximum value of $D_{\overrightarrow{u}}f(x_0,y_0)$ is $||\mathrm{\nabla }f(x_0,y_0)||$.

Example 4: Find the direction for which the directional derivative of $f(x,y)=3x^2+2xy+y^2+5y-11$ at $(-2,3)$ is a maximum. What is the maximum value? 

Exercise 4: Find the direction for which the directional derivative of $f(x,y)=x^2-2xy+3y^2+3y-1$ at $(1,-1)$ is a maximum. What is the maximum value? 

Theorem: Gradient Is Normal to the Level Curve 

Suppose the function $z=f(x,y)$ has continuous first-order partial derivatives in an open disk centered at a point $(x_0,y_0)$. If $\mathrm{\nabla }f(x_0,y_0)\ne 0$, , then $\mathrm{\nabla }f(x_0,y_0)$, is normal to the level curve of $f$ at $(x_0,y_0)$.

Example 5: For the function $f(x,y)=x^2-4x+y^2+2y+4$, find a tangent vector to the level curve at point $(1,2)$. Graph the level curve corresponding to $f(x,y)=9$ and draw in $\mathrm{\nabla }f(1,2)$ and a tangent vector.

Exercise 5: For the function $f(x,y)=x^2+2x+y^2-6y+4$, find a tangent vector to the level curve at point $(2,1)$. Graph the level curve corresponding to $f(x,y)=7$ and draw in $\mathrm{\nabla }f(2,1)$ and a tangent vector.

Definition: Directional Derivative

Suppose $w=f(x,y,z)$ is a function of three variables with a domain of $D$. Let $(x_0,y_0,z_0)$ be a point in the domain of $w$ and $\overrightarrow{u}=<a,b,c>$ be an unit vector. Then the directional derivative of $f$ in the direction of $\overrightarrow{u}$ is given by 

$$D_{\overrightarrow{u}}f(x_0,y_0,z_0)=\underset{h\to 0}{lim}\frac{f(x_0+ah,y_0+bh,z_0+ch)-f(x_0,y_0,z_0)}{h}$$

 provided the limit exists.

Theorem: Directional Derivative of a Function of Three Variables 

Let $w=f(x,y,z)$ be a function of three variables $x$, $y$ and $z$, and assume that $f_x(x,y,z)$, $f_y(x,y,z)$ and $f_z(x,y,z)$ exist. Then the directional derivative of $f$ in the direction of unit vector $\overrightarrow{u}=<a,b.c>$ is given by 

$$D_{\overrightarrow{u}}f(x,y,z)=f_x(x,y,z)a+f_y(x,y,z)b+f_z(x,y,z)c.$$

Example 6: Calculate $D_{\overrightarrow{u}}f(1,-2,1)$ in the direction of $\overrightarrow{u}=<2,3,4>$ for the function $f(x,y,z)=x^{2}y-y^{2}z+2xyz$.

Exercise 6: Calculate $D_{\overrightarrow{u}}f(0,1,2)$ in the direction of $\overrightarrow{u}=<1,2,3>$ for the function $f(x,y,z)=x^{3}y+x{y}^{2}z+2y{z}^2$.

Example 7: The temperature $T$ in a metal sphere is inversely proportional to the distance from the center of the sphere (the origin: $(0,0,0)$). The temperature at point $(1,2,2)$ is $120{}^{\circ }C.$

a. Find the rate of change of the temperature at point $(1,2,2)$ in the direction toward point $(2,1,3)$.

b. Show that, at any point in the sphere, the direction of greatest increase in temperature is given by a vector that points toward the origin.

Group work:

1. The electrical potential (voltage) in a certain region of space is given by the function $V(x,y,z)=5x^2-3xy+xyz.$ 

a. Find the rate of change of the voltage at point $(3,4,5)$ in the direction of the vector $<1,1,-1>$. 

b. In which direction does the voltage change most rapidly at point $(3,4,5)$ ? 

c. What is the maximum rate of change of the voltage at point (3, 4, 5)?

2. Find the gradient vector $f(x,y)=x{e}^y-\text{ln}(x),\text{ at }P(-3,0)$. 

3. Find the gradient vector $f(x,y,z)=x\sqrt{y^2+z^2}$, at $P(-2,-1,-1)$.

4. Find the derivative of the function of $f(x,y)=e^{xy}$ at point $(6,7)$ in the direction the function increases most rapidly. 

5. Find the derivative of the function of $f(x,y)=\text{arctan}(\frac{y}{x})$ at point $(-9,9)$ in the direction the function increases most rapidly. 

6. Find the maximum rate of change of $f=x{e}^{-y}$ at $(-1,0)$ and the direction in which it occurs.

7. Find the maximum rate of change of $f=\text{cos}(3x+2y)$ at $(\frac{\pi }{6},-\frac{\pi }{8})$ and the direction in which it occurs.