# Partial Derivatives Instead of Implicit

• Jameson
In summary, on MathWorld's site they said that -\frac{\partial{y}}{\partial{x}}){_f} = -\frac{(\frac{\partial{f}}{\partial{x}})_{y}}{(\frac{\partial{f}}{\partial{y}})_{x}}So can this method be used instead of implicit differentiation? Will I get the same result? This seems kind of like a parametrics process if this is true.Thanks,Jameson

#### Jameson

Gold Member
MHB
On MathWorld's site, they said that

$$(\frac{\partial{y}}{\partial{x}}){_f} = -\frac{(\frac{\partial{f}}{\partial{x}})_{y}}{(\frac{\partial{f}}{\partial{y}})_{x}}$$

So can this method be used instead of implicit differentiation? Will I get the same result? This seems kind of like a parametrics process if this is true.

Thanks,
Jameson

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Yep,that's the formula from the statement of the theorem of implicit functions.

Daniel.

Is that for f(x,y)? If so isn't the right hand side = -1?

y(f,x)?

for $f(x,y) = 0$

$$df = \frac{\partial{f}}{\partial{x}}dx + \frac{\partial{f}}{\partial{y}}dy = 0$$

$$\frac{\partial{f}}{\partial{x}}dx = -\frac{\partial{f}}{\partial{y}}dy$$

$$\frac{dy}{dx} = -\frac{\frac{\partial{f}}{\partial{x}}}{\frac{\partial{f}}{\partial{y}}}$$

Quetzalcoatl9 I'm not sure if you addressed my question. Isnt there a theorem that says the double partial of a function with respect to alternating variables are equal?

$$f(x,y), f_{xy} = f_{yx}$$ Right? Isn't that what's going on here?

whozum said:
Quetzalcoatl9 I'm not sure if you addressed my question. Isnt there a theorem that says the double partial of a function with respect to alternating variables are equal?

$$f(x,y), f_{xy} = f_{yx}$$ Right? Isn't that what's going on here?

I don't understand what you are asking, whozum...do you mean to ask if partial derivatives commute?

I don't know what you mean by commute, but please verify that

$$For \ f(x,y), \ f_{xy} = f_{yx}$$

If so, then please explain why the right hand side of the original OP's equation isn't just -1

If so, then please explain why the right hand side of the original OP's equation isn't just -1

where'd you get the idea the right side was -1?

$$\left(\frac{\delta f}{\delta x}\right)_y = f_{xy}$$

$$\left(\frac{\delta f}{\delta y}\right)_x = f_{yx}$$

$$f_{yx} = f_{xy}$$

edit: I'm obviously missing something, I just want someone to point out what it is.

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Yes,you are.Missing something,that is.We use this notation

$$\left (\frac{\partial f}{\partial x}\right)_{y}\equiv \frac{\partial f(x,y)}{\partial x}$$

,that is we explain which variables are kept constant during the partial differentiaition,viz. "y" in this case.

This notation,or convention,if u prefer,is very common in thermodynamics.

Daniel.

So that hanging y on the LHS is just to indicate that its constant, then. I had that confused as a derivative.

The notation is excellent in a field like thermo-dynamics, where it is convenient to switch between which quantities are to be regarded as independent quantities and which are to be regarded as dependent quantities.

It is rather redundant in a field where there exisst a "natural" choice of the independent variables.

To give u an idea

$$\frac{\partial F}{\partial V}$$

,where F is the free energy/Helmholtz potential & V is the volume,doesn't mean anything in thermodynamics.

Daniel.

Are you talking to me??

Of course,not.I gave that example to Whozum...

Hmmmmm...:tongue2:

Daniel.

arildno said:
Are you talking to me??
...<in bronx-accented Al Pacino voice> :rofl:

I haven't bothered to get a driving license yet.
Perhaps I should..

## What are partial derivatives?

Partial derivatives are a type of derivative used in multivariate calculus to calculate the rate of change of a function with respect to one of its variables while holding all other variables constant. They are denoted by ∂ and are useful for understanding the behavior of a function with multiple independent variables.

## Why use partial derivatives instead of implicit differentiation?

Partial derivatives are used when a function has more than one independent variable, whereas implicit differentiation is used for functions with a single independent variable. Partial derivatives allow us to analyze the changes in a function due to changes in one variable while holding all others constant, making them more useful for multivariate problems.

## How are partial derivatives calculated?

Partial derivatives are calculated by treating all variables except the one being differentiated with respect to as constants. The derivative is then taken with respect to the variable of interest. For example, to find the partial derivative of a function f(x,y) with respect to x, we treat y as a constant and take the derivative of f(x,y) with respect to x.

## What are the applications of partial derivatives?

Partial derivatives have many applications in fields such as physics, economics, and engineering. They are used to analyze multivariate functions in optimization problems, to understand the behavior of complex systems, and to calculate rates of change in various physical and economic processes.

## Can partial derivatives be used for functions with more than two variables?

Yes, partial derivatives can be extended to functions with any number of independent variables. In this case, the partial derivative of a function with respect to one variable is calculated by treating all other variables as constants and taking the derivative with respect to the variable of interest. This allows for a deeper understanding of the behavior of functions with multiple independent variables.