#6. Find the gradient of f(x,y,z)

Question

anonymous · Answer

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anonymous · Answer

@Jhannybean Thank you for that. I guess what i am mainly having trouble with is the "-3(x^2+y^2)z" Im not sure how to differentiate it

anonymous · Answer

would it be -6xz-0 ? @Jhannybean just barely am learning this and my professors notes are not much help :/

jhannybean · Answer

Ok so taking $\dfrac{d}{dx}$ of that function, you treat $z$ and $y$  as constants. $$\frac{d}{dx}(-3x^2z-3y^2z)= -6xz~\checkmark$$

anonymous · Answer

check! i understand that :)

jhannybean · Answer

Sorry, I wrote the derivatives wrong in my first post. It was meant to be $\dfrac{d}{dx}~,~\dfrac{d}{dy}~,~\dfrac{d}{dz}$

jhannybean · Answer

But now taking the derivative with respectto y, treat x and z as constants. Same when taking the derivative with respect to z.

jhannybean · Answer

I'll help you through the first one

anonymous · Answer

ok so d/dy it would be -6yz ?

jhannybean · Answer

Yep

anonymous · Answer

as for d/dz its tricky b/c of the tan^(-1)xz I'm not sure do you differentiate with respect to x or z?

jhannybean · Answer

z.

anonymous · Answer

why is that?

irishboy123 · Answer

NO, z is not enough.   

for the tan term you will have to apply all of ∂/∂x, ∂/∂y, ∂/∂z as per the del/grad operator <∂/∂x, ∂/∂y, ∂/∂z>, but as there is no y term ∂/∂y comes out as zero. there are x and z terms in there.

assume you know how to do it?  just reverse out from tan and do implicitly.

note too that these are partials and the wrong derivative notation is being used in the above posts.  different notation is used for a reason.

happy to help if you need more.

kainui · Answer

What is f(x,y,z)? Pretend that it represents the temperature at some location (x,y,z) in the room. So if you plug in f(1,1,1) directly you will get the temperature at that location. Now as you're holding your thermometer at that point, let's say you're still cold. Where in the room can you move so that you can get the warmest? That's a direction, right? That sounds like a vector! This is exactly what the gradient is. It creates a vector field out of a scalar field and that vector field tells you the direction of greatest increase.