Understanding Total and Partial Derivatives in Multivariable Calculus

In summary, the equalities between total and partial derivatives are true if \frac{dy}{dx}=f(x,y). The first equality is always true, while the second one may be true depending on the specific function.
  • #1
Ted123
446
0
Are the following equalities between total and partial derivatives true if [itex]\frac{dy}{dx}=f(x,y)[/itex]? [tex]\displaystyle \frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} f(x,y)[/tex] [tex]\displaystyle \frac{d^2f}{dx^2} = \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} + \left( \frac{\partial f}{\partial y} \right) ^2 f(x,y)[/tex]
 
Physics news on Phys.org
  • #2
Ted123 said:
Are the following equalities between total and partial derivatives true if [itex]\frac{dy}{dx}=f(x,y)[/itex]? [tex]\displaystyle \frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} f(x,y)[/tex] [tex]\displaystyle \frac{d^2f}{dx^2} = \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} + \left( \frac{\partial f}{\partial y} \right) ^2 f(x,y)[/tex]

This is a good resource
http://en.wikipedia.org/wiki/Total_derivative
 
  • #3

1. What is the definition of a total derivative?

The total derivative of a multivariable function f(x,y) is the rate of change of the function with respect to all of its variables. It takes into account the effects of changes in both the independent and dependent variables.

2. How do you calculate a partial derivative?

To calculate a partial derivative of a multivariable function, you hold all other variables constant and take the derivative with respect to the variable of interest. This results in a new function that represents the rate of change of the original function with respect to that specific variable.

3. What is the difference between a total derivative and a partial derivative?

A total derivative considers the effects of changes in all variables, while a partial derivative only considers the effects of changes in one variable while holding all others constant.

4. What is the chain rule for partial derivatives?

The chain rule for partial derivatives states that to find the derivative of a composite function, you take the partial derivative of the outer function with respect to the inner function, then multiply it by the partial derivative of the inner function with respect to the variable of interest.

5. How are partial derivatives used in real-world applications?

Partial derivatives are used in many fields of science and engineering, such as physics and economics, to model and analyze relationships between multiple variables. They are particularly useful in optimization problems, where finding the maximum or minimum value of a function depends on the values of multiple variables.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
706
  • Calculus and Beyond Homework Help
Replies
5
Views
559
  • Calculus and Beyond Homework Help
Replies
4
Views
635
Replies
4
Views
603
  • Calculus and Beyond Homework Help
Replies
6
Views
793
  • Calculus and Beyond Homework Help
Replies
6
Views
505
  • Calculus and Beyond Homework Help
Replies
3
Views
720
  • Calculus and Beyond Homework Help
Replies
2
Views
403
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Back
Top