# Partial Differentiation Question

#### Psycopathak

I have a basic question about taking partial derivatives.

Say I have a function of 3 variables and i want the derivative of only one. Do I take the derivative of the one variable and HOLD THE OTHER TWO CONSTANT? Or, do I take the derivative of the variable and TREAT THE OTHER TWO AS CONSTANTS?

For example

If I have f(x,y,z) = xy2+z3 and I want to find fx

Does that mean I get 2xy or 2xy+z3?

Like does that z drop out or do I literally not touch it? This isn't a homework question. I'm just trying to understand what the rule is.

#### sutupidmath

well, what is the definition of a partial derivative of a function of several variables? What is the rule of taking the derivative with respect to a specific variaable?

2xy is your answer. Although i initially didn't know the difference between:"HOLD THE OTHER TWO CONSTANT" vs "TREAT THE OTHER TWO AS CONSTANTS", i believe, the second one applies.

You have to treat the other variables as constants.

A side note: if we have a function of two variables, say, z=f(x,y), and we want to take the partial derivative with respect to x, fx, then geometrically what it means is that we are finding the slope of the tangent lines at any point x on the traces Ci in the vertical planes y=yo parallel to the xz plane. What this means is tha on background we are treating y as a constant.

#### jacophile

$$y^{2}$$

#### sutupidmath

$$y^{2}$$
Well, if you are taking the partial with respec to x, then yes. But if you are taking the partial with respec to y, then it is 2xy.

#### jacophile

I want to find fx
Sorry, I guess I have the notation confused, I thought the fx was the derivative wrt x.

#### sutupidmath

Sorry, I guess I have the notation confused, I thought the fx was the derivative wrt x.
No,you are correct $$f_x$$ is the partial with respect to x.

#### jacophile

If I have f(x,y,z) = xy2+z3 and I want to find fx

Does that mean I get 2xy or 2xy+z3?
Um, so the answer to the original question is neither. You get y2.

#### sutupidmath

Um, so the answer to the original question is neither. You get y2.
OOO, i just saw that he was differentiating wrt to x. Yep, that is the correct answer.

#### HallsofIvy

I have a basic question about taking partial derivatives.

Say I have a function of 3 variables and i want the derivative of only one. Do I take the derivative of the one variable and HOLD THE OTHER TWO CONSTANT? Or, do I take the derivative of the variable and TREAT THE OTHER TWO AS CONSTANTS?
What do you understand as the difference between "HOLD THE OTHER TWO CONSTANT" and "TREAT THE OTHER TWO AS CONSTANTS"?

For example

If I have f(x,y,z) = xy2+z3 and I want to find fx

Does that mean I get 2xy or 2xy+z3?

Like does that z drop out or do I literally not touch it? This isn't a homework question. I'm just trying to understand what the rule is.
Neither as you have been told. Whether you "TREAT THE OTHER TWO AS CONSTANTS" or
"HOLD THE OTHER TWO CONSTANT", what is the derivative of f(x)= xb2+ c, where b and c are constants.

#### Psycopathak

No,you are correct $$f_x$$ is the partial with respect to x.

#### Psycopathak

What do you understand as the difference between "HOLD THE OTHER TWO CONSTANT" and "TREAT THE OTHER TWO AS CONSTANTS"?

Neither as you have been told. Whether you "TREAT THE OTHER TWO AS CONSTANTS" or
"HOLD THE OTHER TWO CONSTANT", what is the derivative of f(x)= xb2+ c, where b and c are constants.
f'(x) = 2xb

Yeah I geometrically understand what it is to find the partial derivative. Basically you're intersecting a plane with a surface and finding the derivative of the curve that the intersection of the plane and the surface makes.

But I was mistaken earlier.

f(x,y,z) = x3yz+3yz-2y+z and I wanted fx Do I get

fx(x,y,z) = 3x2yz
OR
fx(x,y,z) = 3x2yz +3yz-2y+z

I guess the first one is correct because since the rest are treated as constants, they simply drop out.

Also, how would I do something like say:

f(x,y,z) = e(x2-y2+z2), find fx

I'm just trying to understand the rules of partial differentiation. Thanks for your help so far!

#### sutupidmath

f(x,y,z) = e(x2-y2+z2), find fx

I'm just trying to understand the rules of partial differentiation. Thanks for your help so far!
Same thing as above, just that in this case you need to apply chain rule! THat is, you still are considering y,z as constants, but the extra twist here is that you have a function,namely e, raised to a function(which is not simply an independent variable), so as i mentioned, you'll need chain rule here.

#### HallsofIvy

f'(x) = 2xb

Yeah I geometrically understand what it is to find the partial derivative. Basically you're intersecting a plane with a surface and finding the derivative of the curve that the intersection of the plane and the surface makes.

But I was mistaken earlier.

f(x,y,z) = x3yz+3yz-2y+z and I wanted fx Do I get

fx(x,y,z) = 3x2yz
OR
fx(x,y,z) = 3x2yz +3yz-2y+z

I guess the first one is correct because since the rest are treated as constants, they simply drop out.

Also, how would I do something like say:

f(x,y,z) = e(x2-y2+z2), find fx

I'm just trying to understand the rules of partial differentiation. Thanks for your help so far!
What is the derivative of
$$f(x)= e^{x^2+ a}$$?

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