Derivative Distribution in Multivariable Calculus

[Dawei]

I'm trying to simplify this. I have two functions, f₀(z), and f'(t, x, y, z). One is the 'base' value that varies only with height, the other is the small 'perturbation' value that varies with all four variables.

I am substituting these into an equation that calls for ∂(f)/∂(x)

Do I write this as ∂ (f₀ + f' ) / ∂(x) ?

And if so, how can I simplify it? Since f₀ is not a function of x, can I just cross it out completely? Is it equal to ∂(f₀) / ∂(x) + ∂(f') / ∂(x) ?

 

[Jerbearrrrrr]

What's "f"?
[edit]
Oh, f is the sum of the two things.
Oops.

And yes, you're right.
Derivative of a sum equals the sum of derivatives.
And yes, ∂(f0) / ∂(x)=0.

 

[Dawei]

In this case it is density...

By the way I'd also like to know the general answer here. That is, if the f0 term were a function of x as well.

 

[Jerbearrrrrr]

Sorry, I didn't pick up that f:=f0 + f '
somehow. lol

 

[Mute]

In this case it is density...

By the way I'd also like to know the general answer here. That is, if the f0 term were a function of x as well.

Yes. The derivative is a linear operation, so it distributes. You can see this from the definition. If a function f depends on coordinates x_0,x_1,...,x_n, then the partial derivative of f with respect to one of the x's, x_i for generality, is

$$\lim_{h \rightarrow 0} \frac{f(x_0,x_1,\dots,x_i+h,\dots,x_n) - f(x_0,x_1,\dots,x_i,\dots,x_n)}{h} = \frac{\partial f(x_0,x_1,\dots,x_i,\dots,x_n)}{\partial x_i}$$

If you plug in f(x_0,x_1,...,x_n) = f_0(x_0,x_1,...,x_n) + f_1(x_0,x_1,...,x_n), you can rearrange that so that you get the sum of the derivatives. Assuming that f_0 depends only on x_j (for j not equal to i) causes that term to drop out.