Need help on a partial derivative problem

In summary: So you need to differentiate:f_x=3x^2+2xy+1 w.r.t. y. f_{xy}=2x So, f_{xy} is not equal to f_{yx}.However, in this case, when you differentiate f with respect to x and then y, or vice versa, you will get the same result. This is because the function is smooth and the order of differentiation does not matter. But in general, for non-smooth functions, the mixed partial derivatives may not be equal. In summary, the mixed partial derivatives of the given function f(x,y)=x^3+x^2y+x+4 are f_{xy}=2x and f_{yx}=2x.
  • #1
jzq
55
0
Find the second-order partial derivatives of the given function. In each case, show that the mixed partial derivatives [tex]f_{xy}[/tex] and [tex]f_{yx}[/tex] are equal.

Function:
[tex]f(x,y)=x^{3}+x^{2}y+x+4[/tex]

My work (Correct me if I am wrong):
[tex]\frac{\partial{f}}{\partial{x}}}=3x^{2}+2xy+1[/tex]

[tex]\frac{\partial{f}}{\partial{y}}}=x^{2}[/tex]

[tex]f_{xx}=6x+2y[/tex]

[tex]f_{yy}=0[/tex]

[tex]f_{xy}=6x+2y[/tex]

[tex]f_{yx}=0[/tex]

If I am correct, which I am probably not, how could [tex]f_{xy}[/tex] possibly be equal to [tex]f_{yx}[/tex]? Shouldn't that always be true anyways? If that's so, then obviously I messed up somewhere. Please help!
 
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  • #2
How did you find those mixed partials? You seem to have done the exact same thing to find [itex]f_{xy}[/itex] as you did for [itex]f_{xx}[/itex] (and the same for [itex]yy[/itex] and [itex]yx[/itex]). I think if you check your work over, you'll see that you differentiated wrt the wrong variables a couple of times :wink:
 
  • #3
Use Jacobi's notation for partial derivatives.It will leave no room for any confusion once u realize the order of differentiation.And if u use Lagrange's one,do it properly

[tex] \frac{\partial f}{\partial x}\equiv f'_{x} [/tex]

Daniel.
 
  • #4
Nothing wrong with notation evolving. I've never seen notation like [itex]f^\prime_x[/itex], though.
 
  • #5
Did I atleast get the first partial derivatives correct?
 
  • #6
Your second partials are wrt to the wrong variables

[tex] f_{xy} [/tex] means differentiate [itex] f_x [/itex] with respect to y.
 

Related to Need help on a partial derivative problem

1. What is a partial derivative?

A partial derivative is a mathematical concept used to determine the rate of change of a multivariable function with respect to one of its variables, while holding all other variables constant.

2. How do I solve a partial derivative problem?

To solve a partial derivative problem, you will need to use the appropriate partial derivative formula and apply it to the given function. You will also need to carefully identify the variable you are taking the derivative with respect to and treat all other variables as constants.

3. What is the purpose of using partial derivatives?

Partial derivatives are useful in many areas of science, engineering, and economics. They help to describe the behavior of a function in multiple dimensions and are used in optimization problems, curve fitting, and in the study of rates of change.

4. Can you provide an example of a partial derivative problem?

Sure! An example of a partial derivative problem could be finding the partial derivative of the function f(x,y) = x^2y + 3xy^2 with respect to x. Using the appropriate formula, we would get fx(x,y) = 2xy + 3y^2.

5. Are there any important rules or properties to keep in mind when working with partial derivatives?

Yes, there are a few important rules to remember when working with partial derivatives. These include the chain rule, product rule, and quotient rule. It is also important to remember that the order in which you take the partial derivatives can affect the final result.

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