How Do You Calculate the Gradient of Multivariable Functions?

[vorcil]

Homework Statement

Find the gradients of the following functions:

When I say gradient, I'm not just differentiating the functions, apparently I have to do it this way (because it's in my physics book)

f(x,y,z) = x^2 + y^3 + z^4
f(x,y,z) = x^2 y^3 z^4
f(x,y,z) = e^x sin(y) ln(z)

Homework Equations

$$\nabla {\bf r} = \frac{\partial r}{\partial x} \hat{x} + \frac{\partial r}{\partial y} \hat{y} + \frac{\partial r}{\partial z} \hat{z}$$

I'm not just differentiating the functions

where $$\frac{r}{|r|}$$ = gradient

The Attempt at a Solution

for the first question, finding the gradient of
f(x,y,z) = x^2 + y^3 + z^4

$$\nabla {\bf f(x,y,z) } = \frac{\partial r}{\partial x} \hat{x} + \frac{\partial r}{\partial y} \hat{y} + \frac{\partial r}{\partial z} \hat{z}$$

differentiating partially,$$\nabla$$
$$\nabla<br /> (x^2 + y^3 + z^4) = 2x \hat{x} + 3y^2 \hat{y} + 4z^3 \hat{z}$$

finding the magnitude of the function
$$\sqrt{(x^2)^2 + (y^3)^2 + (z^4)^2} = \sqrt{x^4 + y^6 + z^8}$$

using the formula to find the gradient,
$$\frac{r}{|r|} = \frac{\nabla (x^2 + y^3 + z^4) }{\sqrt{x^4 + y^6 + z^8}} = \frac{2x \hat{x} + 3y^2 \hat{y} + 4z^3 \hat{z}}{\sqrt{x^4 + y^6 + z^8}}$$

not sure if this is right or not, also I don't know if I can simplifier it further and if this satisfies as an answer,
my book has no answers in the back to check, and solutions manuals are impossible to get to students, and the only way to find out if I'm right is to either post it here, or ask my lecturer, and I feel that this is a question that most people should be able to do, so it's embarrassing for me to ask my lecturer at tutorials

 

[vorcil]

[edit] fixed the latex

 

[lanedance]

i may be missing something, but why not just use the cartesian form of the gradient

$$\nabla = \begin{pmatrix} <br /> \frac{\partial }{\partial x} \\ <br /> \frac{\partial }{\partial y} \\ <br /> \frac{\partial }{\partial z} <br /> \end{pmatrix}$$