Partial Differentiation Question

In summary, when taking a partial derivative of a function with multiple variables, one can either treat the other variables as constants or hold them constant. In the example of f(x,y,z) = xy2+z3, finding fx would result in 2xy. This means that the other variables, y and z, are treated as constants. For a more complex function like f(x,y,z) = e(x2-y2+z2), finding fx would involve using the chain rule, where the other variables are still treated as constants.
  • #1
Psycopathak
13
0
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.
 
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  • #2
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.
 
  • #3
Huh? Isn't the answer

[tex]y^{2}[/tex]
 
  • #4
jacophile said:
Huh? Isn't the answer

[tex]y^{2}[/tex]

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.
 
  • #5
Psycopathak said:
I want to find fx

Sorry, I guess I have the notation confused, I thought the fx was the derivative wrt x.
 
  • #6
jacophile said:
Sorry, I guess I have the notation confused, I thought the fx was the derivative wrt x.

No,you are correct [tex]f_x[/tex] is the partial with respect to x.
 
  • #7
Psycopathak said:
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.
 
  • #8
jacophile said:
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.
 
  • #9
Psycopathak said:
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.
 
  • #10
sutupidmath said:
No,you are correct [tex]f_x[/tex] is the partial with respect to x.

Yeah I made a mistake, sorry about that.
 
  • #11
HallsofIvy said:
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.

Like If I had

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!
 
  • #12
Psycopathak said:
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.
 
  • #13
Psycopathak said:
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.

Like If I had

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
[tex] f(x)= e^{x^2+ a}[/tex]?
 

1. What is partial differentiation?

Partial differentiation is a mathematical concept that involves finding the rate of change of a multivariable function with respect to one of its variables, while holding all other variables constant. It is used in calculus and is an important tool in many fields of science, such as physics, economics, and engineering.

2. Why do we use partial differentiation?

Partial differentiation allows us to analyze how a function changes as one of its variables changes, while keeping other variables constant. This is useful in situations where multiple variables are involved and we want to understand the relationship between them. It also helps us optimize functions by finding the points where the rate of change is zero.

3. How is partial differentiation different from ordinary differentiation?

In ordinary differentiation, we find the rate of change of a function with respect to one variable. In partial differentiation, we find the rate of change of a multivariable function with respect to one variable while holding all other variables constant. This means that in partial differentiation, we are only focusing on the impact of one variable on the function, rather than considering all variables at once.

4. What are the applications of partial differentiation?

Partial differentiation is used in many areas of science, including physics, economics, engineering, and statistics. It is used to analyze the behavior of systems with multiple variables, to optimize functions, and to solve problems involving rates of change. For example, partial differentiation is used in economics to find the marginal rate of substitution, and in physics to calculate the change in temperature over time.

5. What are some common techniques for solving partial differentiation problems?

There are several techniques for solving partial differentiation problems, including the method of substitution, the method of implicit differentiation, and the method of logarithmic differentiation. These techniques involve manipulating the given function and using rules of differentiation to find the partial derivatives. It is important to have a strong understanding of basic calculus concepts and rules in order to effectively solve partial differentiation problems.

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