Find the directional derivative of $$f(x,y,z) = yz + x^4$$ at (2,3,1) in the direction of vector v=i+j+k.

I have gotten as far as finding the directional derivative (gradient) at (2,3,1) which is:

32i+3j+k

I dont know how to do the second part of this problem. What does it mean to find this direction? What do I have to do to the gradient to find it? My book is very vague, please help!

Question

Find the directional derivative of $$f(x,y,z) = yz + x^4$$ at (2,3,1) in the direction of vector v=i+j+k.

I have gotten as far as finding the directional derivative (gradient) at (2,3,1) which is:

32i+3j+k

I dont know how to do the second part of this problem. What does it mean to find this direction? What do I have to do to the gradient to find it? My book is very vague, please help!

anonymous · Answer

Tips?

anonymous · Answer

The box for the answer says fu= _____ , where u looks like a unit vector, is that what this is asking for? A unit vector of the gradient?

anonymous · Answer

Never mind, that, I tried, and was told my answer should be a number, not a vector. Guess I misinterpreted that.

anonymous · Answer

attachment

unklerhaukus · Answer

$$
\newcommand	\ve	[1]	{	\mathbf{#1}		}% vector
\nabla_{\ve v}f(\ve x)=\nabla f(\ve x)\cdot{\ve v}\
\
\
f=yz+x^4\
\ve v=(1,1,1)$$

anonymous · Answer

I think I understand what you wrote, but it still isnt clear to me what "in the direction of" means... I can find the gradient just fine, I've just no idea what to do with it.

unklerhaukus · Answer

once you have found the gradient, take the dot product with the direction vector

anonymous · Answer

ahhhhh, that makes sense, thank you!