Calculate partial derivatives and mixed partial derivatives

[dyn]

Hi. I know how to calculate partial derivatives and mixed partial derivatives such as ∂²f/∂x∂y but I've now become confused about something. If I have a function of 3 variables eg. f(x,y,z) and I calculate ∂_x then I am differentiating wrt x while holding y and z constant. Does that mean ∂_x then becomes a function of x only ie does ∂_x f(x,y,z) = φ( x ) ? If it does then ∂_y and ∂_z will always be zero but I know this is not the case. I'm confused !

 

[fresh_42]

Hi. I know how to calculate partial derivatives and mixed partial derivatives such as ∂²f/∂x∂y but I've now become confused about something. If I have a function of 3 variables eg. f(x,y,z) and I calculate ∂_x then I am differentiating wrt x while holding y and z constant. Does that mean ∂_x then becomes a function of x only ie does ∂_x f(x,y,z) = φ( x ) ? If it does then ∂_y and ∂_z will always be zero but I know this is not the case. I'm confused !

No. It's still a function of three variables.
Differentiating along a single variable simply means to consider the change in values along this coordinate.

You might want to play around a little with Wolfram, e.g. http://www.wolframalpha.com/input/?i=f(x,y)+=+xy^3++4+x^2
Imagine the partial derivation $\partial_x$ as a tangent in $x-$direction. It still varies with $y$ and $z$.

 

[dyn]

Thanks. So it is incorrect to write ∂_x f(x, y ,z) = g ( x ) ?

 

[FactChecker]

Thanks. So it is incorrect to write ∂_x f(x, y ,z) = g ( x ) ?

Unless the variables y and z disappear as a result of differentiating wrt x, they are still there. They might disappear, but don't count on it.

 

[fresh_42]

Thanks. So it is incorrect to write ∂_x f(x, y ,z) = g ( x ) ?

Yes.

You could of course evaluate the derivative at some point $p=(x_0,y_0,z_0)$ and get
$$\frac{\partial}{\partial_x}\bigg{|}_p f(x,y,z) = g(x_0,y_0,z_0)$$
Or if you want to examine the $x-$component, you could consider $g(x,y_0,z_0)$ and get a function in one variable, because you fixed $y=y_0$ and $z=z_0$. But this is another issue.