Taking Differentials to Find Partial Derivative

In summary, the conversation is about taking differentials of a given equation in order to find the partial derivative \frac{{\partial p}}{{\partial V}}. The person asking for help is unsure how to deal with the brackets and is seeking hints. Another person suggests using the product rule for differentials and explains how to use it. However, the first person is getting unexpected results and is unsure why they cannot rearrange the equation. The second person clarifies that rearranging is necessary in order to solve for p in terms of V, which is required to find the partial derivative.
  • #1
jamjar
10
0
Hi,
I'm trying to take differentials of the following equation

[tex](p + \frac{a}{{V^2 }})(V - b) = C[/tex]

in order to find the partial derivative [tex]\frac{{\partial p}}{{\partial V}}[/tex]

I know there's an easier way to do it but I have to take differentials.
I'm just not sure how to deal with the brackets without multiplying out (I can't rearrange the equation).

Any hints welcome.
 
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  • #2
Use the product rule!
 
  • #3
How do I do that with differentials?
I end up with crazy results.
 
  • #4
You "can't" rearrange the equation, as in the problem won't let you? The product rule for differentials is [tex] d(fg) = g df + f dg [/tex].
 
  • #5
If I use that product rule I end up with a free floating [tex]p[/tex] in my equation where I know that [tex]\frac{{\partial p}}{{\partial V}}[/tex] doesn't have a term in [tex]p[/tex]
 
  • #6
Yes, you then have to solve for p in terms of V using the original equation. This I why I don't undestand why you can't just solve for p in terms of V from the start since you have do it eventually anyway.
 

1. What is the purpose of taking differentials to find partial derivatives?

The purpose of taking differentials to find partial derivatives is to understand how a function changes with respect to one of its variables while holding all other variables constant. This is useful in many fields of science, including physics, engineering, and economics.

2. How do you take differentials to find partial derivatives?

To take differentials to find partial derivatives, you first need to differentiate the function with respect to the variable you want to take the partial derivative of. Then, you multiply that derivative by the differential of the variable. This process is known as the chain rule.

3. What is the difference between partial derivatives and ordinary derivatives?

The main difference between partial derivatives and ordinary derivatives is that partial derivatives involve functions with multiple variables, while ordinary derivatives involve functions with only one variable. Partial derivatives also hold all other variables constant, while ordinary derivatives do not.

4. Can you give an example of taking differentials to find partial derivatives?

Sure! Let's say we have a function f(x,y) = x^2 + y^3. To find the partial derivative of f with respect to x, we would first find the derivative of x^2, which is 2x. Then, we would multiply that by the differential of x, which is dx. So the partial derivative of f with respect to x would be 2x dx.

5. Why are partial derivatives important in science?

Partial derivatives are important in science because they allow us to understand the behavior of complex systems with multiple variables. They are used in many fields, such as physics, economics, and engineering, to analyze how a system changes and to make predictions based on those changes.

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