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## Homework Statement

(a) Find the directional derivative of

*z*=

*x*2

*y*at (3,4) in the direction of 3π/4 with the

*x*-axis. Give an exact answer.

(b) Find the directional derivative of

*z*=

*x*2

*y*at (3,4) in the direction that makes an angle of 3π/4 with the gradient vector at (3,4). Give an exact answer.

(c) In which direction is the directional derivative the largest? Give your answer as a vector.

## Homework Equations

##F\vec u (a,b) = \nabla F(a,b) \cdot \vec u##

Found the following two equations using the one above:

##\nabla F(3,4) = \langle 24,9 \rangle##

##\frac { \vec \nabla F(3,4)} {\left\|\vec \nabla F(3,4)\right\|} = \langle \frac {8}{\sqrt{73}},\frac {3} {\sqrt {73}} \rangle ##

## The Attempt at a Solution

A. I found the directional derivative as ##\frac {-15}{\sqrt{2}}##.

B. I'm stuck here. I can't seem to figure out how to get the unit vector I need to dot the gradient with.

I've tried using the geometric and algebraic definitions of the dot product and the fact that##\sqrt{{u_1}^2 + {u_2}^2} = 1## to find each component of u, but I'm just getting garbage as an answer (1 = not 1 ). Part of the problem is that I'm supposed to give an exact answer, which I haven't quite figured out how to do yet.

I just tried to use some trig logic to get the angles I need, but it doesn't seem to be working either.

Since ##θ = arccos (x)##, and ##θ = arcsin (y)##, I set my unit vector to ##\vec u = \langle cos(\frac {3π}{4} + arccos\frac {8}{\sqrt{73}}), sin(\frac {3π}{4} + arcsin\frac {3}{\sqrt{73}})\rangle## and dotted it with the gradient, but that's wrong as well.

C. Haven't gotten to this part yet.

The good news is that after 6 years on PF, I've finally learned the basics of LaTeX!

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