First and second order partial derivatives

In summary, Sean Feely is asking for help with homework and states that he is not really sure how to get started. He asks for the first and second order partial derivatives of f with respect to x and y respectively. He calculates the derivatives for x as if y was a constant and then does the same for y. He also mentions that it is okay to take derivatives with respect to x first and y second, y first x second, or y first and x second.
  • #1
feely
24
0
Hello,

I was wondering if I could get some help with a question I have.

Homework Statement



We are asked to find the first and second order partial derivatives of

f(x,y) = x^2 - y^2 - 4x^2/(y - 1)^2 (sorry, I don't know how to write this in latex).

I am not really sure how to get started with this, so any help at all would be fantastic.

Thanks

Sean Feely
 
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  • #2
Just calculate the derivatives in respect to one variable while treating the other variable as if it was a constant. Do this for both variables in different orders and you should get 6 derivatives (2 of the 1st order and 4 of the 2nd order) although two of them will be the same.
 
  • #3
feely said:
We are asked to find the first and second order partial derivatives of

f(x,y) = x^2 - y^2 - 4x^2/(y - 1)^2 (sorry, I don't know how to write this in latex).

First and second order partial derivatives with respect to which variable? Both?
 
  • #4
You are being asked to find [tex]\frac{{\partial f(x,y)}}{{\partial x}},\frac{{\partial f(x,y)}}{y},\frac{{\partial ^2 f(x,y)}}{{\partial x^2 }},\frac{{\partial ^2 f(x,y)}}{{\partial y^2 }}[/tex] and possibly the mixed partial derivative, [tex]\frac{{\partial f(x,y)}}{{\partial x\partial y}}[/tex].

Let's say you have a function [tex]f(x,y) = 3x + 2y + 6xy[/tex]. To find the first partial with respect to x, that is [tex]\frac{{\partial f(x,y)}}{{\partial x}}[/tex], you simply pretend y is a constant. Thus, you find the partial with respect to x is [tex]\frac{{\partial f(x,y)}}{{\partial x}} = 3 + 6y[/tex]. Since y is constant, 2y is a constant with respect to x, and thus it goes away when you differentiate with respect to x. The second derivative from there is obviously 0 since the derivative of 3 is 0 and the derivative of 3y is 0 because y is a constant just like the number 3. When you take partials with respect to y, you do the same thing except with x as constant. The first derivative with respect to y would be 2 + 6x and again, the second derivative would be 0.

The mixed partial is a bit different as you switch the order. Either way is fine but you can first take the derivative with respect to x which gives you 3 + 6y. Then you have to take the derivative of y, and since y is no longer a constant, you see that the mixed partial is simply equal to 6. It actually does not matter which way you take your mixed partials, x first y second, y first x second in most cases.
 

1. What is the difference between first and second order partial derivatives?

The first order partial derivative is a measure of how a function changes with respect to one independent variable, while holding all other independent variables constant. The second order partial derivative is a measure of how the first order partial derivative changes with respect to that same independent variable.

2. How do you find the first and second order partial derivatives?

To find the first order partial derivative, you take the derivative of the function with respect to the desired independent variable, treating all other independent variables as constants. To find the second order partial derivative, you take the derivative of the first order partial derivative with respect to the same independent variable.

3. What is the chain rule in relation to first and second order partial derivatives?

The chain rule is a method for finding the derivative of a composite function. In the context of first and second order partial derivatives, the chain rule is used to find the derivative of a function with respect to one independent variable when it is dependent on multiple other independent variables.

4. Can first and second order partial derivatives be used to determine the slope of a function?

Yes, the first order partial derivative can be used to determine the slope of a function in a particular direction, while the second order partial derivative can be used to determine the rate of change of that slope.

5. Are first and second order partial derivatives used in real-world applications?

Yes, first and second order partial derivatives are used in many real-world applications, particularly in fields such as physics, economics, and engineering. They are used to model and analyze relationships between multiple variables and to make predictions about how a system will behave under different conditions.

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