Find the Four 2nd Partial Derivatives of f(x,y)

Zx = ye^x + cosx + xycosxIn summary, the four second partial derivatives for f(x,y) = y^2e^x + xycosx are Zxx = y^2e^x - ycosx, Zxy = ye^x + cosx + xycosx, Zyy = 2e^x + xcosx, and Zyx = ye^x + cosx + xycosx. To solve for Zxy and Zyx, remember to treat the other variable as a constant when differentiating.
  • #1
p4nda
16
0
Find the four second partial derivatives for f(x,y) = y^2e^x + xycosx

I am stuck on the last part... here's what I got so far:

Zx = y^2e^x - ysinx
Zxx = y^2e^x - ycosx
Zxy = ?

Zy = 2ye^x + xcosx
Zyy = 2e^x + xcosx
Zyx = ?


I need help with solving for xy. Both should end up being the same. :eek:
 
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  • #2
the trick is to ignore the other variable whenever you are not differentiating with respect to it. eg.
[tex]\frac{\partial}{\partial x} x e^y = e^y[/tex] and
[tex]\frac{\partial}{\partial y} x e^y = x e^y[/tex]

in first case you treat y as if it is a constant whereas in the second you treat x as constant instead. the same rules apply when you go to higher derviatives
eg.
[tex]\frac{\partial}{\partial x} \frac{\partial}{\partial y} (x e^y)
= \frac{\partial}{\partial x}(x e^y) = e^y[/tex]
 
  • #3
by the way your Zx is wrong
 

FAQ: Find the Four 2nd Partial Derivatives of f(x,y)

What is the purpose of finding the four 2nd partial derivatives of f(x,y)?

The purpose of finding the four 2nd partial derivatives of f(x,y) is to determine the rate of change of a multivariable function with respect to each of its two variables. This can help in understanding the behavior of the function and its critical points.

How do you calculate the 2nd partial derivatives of f(x,y)?

To calculate the 2nd partial derivatives of f(x,y), you must first find the 1st partial derivatives with respect to both variables. Then, you take the partial derivative of each of these results with respect to the same variable again. This will give you four 2nd partial derivatives.

What do the four 2nd partial derivatives represent?

The four 2nd partial derivatives represent the rate of change of the slope of the original function in four different directions. Specifically, they represent the curvature of the function in the x-direction, y-direction, and two diagonal directions.

How do the four 2nd partial derivatives relate to each other?

The four 2nd partial derivatives are related in that they are all second-order derivatives of the original function. This means that they are all measuring the curvature of the function in different directions.

What can the four 2nd partial derivatives tell us about the behavior of the function?

The four 2nd partial derivatives can give us information about the local extrema of the function, as well as the concavity and convexity of the function in different directions. They can also help in determining the type of critical point (maximum, minimum, or saddle point) of the function.

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