# Directional Derivative

1. Feb 21, 2016

### RyanTAsher

1. The problem statement, all variables and given/known data

Find the directional derivative of $f$ at $P$ in the direction of $a$.

$f(x,y) = 2x^3y^3 ; P(3,4) ; a = 3i - 4j$

2. Relevant equations

$D_u f(x_0, y_0, z_0) = f_x(x_0, y_0, z_0)u_1 + f_y(x_0, y_0, z_0)u_2$

3. The attempt at a solution

$f_x (x,y) = 6x^2y^3$
$f_y (x,y) = 6x^3y^2$

$f_x (3,4) = 3456$
$f_y (3,4) = 2592$

$D_u f(x_0, y_0) = 3456u_1 +2592 u_2$

$u = \frac {a} {||a||} = \frac {\langle 3,4 \rangle} {5} = \langle \frac {3} {5}, \frac {4} {5} \rangle$

$D_u f(x_0, y_0) = 3456(\frac {3} {5}) + 2592(\frac {4} {5})$

$D_u f(x_0, y_0) = \frac {20736} {5}$

Now, my program wants this an exact number, no tolerance. It won't accept division either, so I don't know how to put in 20736/5. Just wondering if I made a mishap somewhere within the solution.

2. Feb 21, 2016

### SteamKing

Staff Emeritus
You can't enter in a decimal number?

3. Feb 21, 2016

### RyanTAsher

Wouldn't that not be exact, but approximate form though? Or if I decimal isn't repeating is it considered exact?

4. Feb 21, 2016

### SteamKing

Staff Emeritus
So, you're saying that (1/2) = 0.5 is only an approximation and not an exact representation? Interesting.

5. Feb 21, 2016

### RyanTAsher

So, I'm guessing it's not an approximation? Makes sense, good to learn something new. I will attempt to insert my answer.

6. Feb 21, 2016

### RyanTAsher

I attempted the answer of 4147.2, and it was incorrect. Therefore, my work must be incorrect somewhere.

7. Feb 21, 2016

### SteamKing

Staff Emeritus
What if the wrong answer has been programmed into the software you're using?

8. Feb 21, 2016

### RyanTAsher

I've spoken to other students who have the same problem, but with different numbers, so I'm pretty positive that the programs solution is correct, but now that I am at home I don't have access to see any of their solutions.

9. Feb 21, 2016

### Ray Vickson

$u_y \neq 4/5$; go back and check your work.

10. Feb 21, 2016

### RyanTAsher

Ah, thank you I missed that. I have remodeled my work, and the solution turns out to be 0, and is correct. Thank you both for your time.