Understanding Mixed Partial Derivatives: How Do You Solve Them?

In summary, while working at home during the COVID-19 pandemic, I took to seeing if I can still do math from undergrad (something I do once in a while to at least pretend my life isn't dominated by excel). I then reviewed partial derivatives (which I haven't really thought about in a good seven-ish years), but got stuck on the mixed derivative. After figuring out that I was treating y as a constant, I was able to solve for the cosine and sinine of xy.
  • #1
atomicpedals
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While working at home during the COVID-19 pandemic I've taken to seeing if I can still do math from undergrad (something I do once in a while to at least pretend my life isn't dominated by excel). So to that I've been reviewing partial derivatives (which I haven't really thought about in a good seven-ish years).

The exercise I'm working asks the following: Compute all first and second partial derivatives, including mixed derivatives, of the following ##0 = sin(xy) - x^2 -y^2##.

I think I'm good on the first and second partials:
$$\frac {\partial V}{\partial x} = y cos(xy) - 2x$$
$$\frac {\partial V}{\partial y} = x cos(xy) - 2y$$
$$\frac {\partial^2 V}{\partial x^2} = -y^2 sin(xy) - 2$$
$$\frac {\partial^2 V}{\partial y^2} = -x^2 sin(xy) - 2$$
Where I get tripped up is on the mixed derivative:
$$\frac {\partial^2 V}{\partial x \partial y} = \frac {\partial}{\partial x} \left( \frac {\partial V}{\partial y} \right) = \frac {\partial}{\partial x} \left( x cos(xy) - 2y \right)...$$
And this is where I get hung up. Do I simply proceed with an application of the production rule, ##(f \cdot g)' = f' \cdot g + f \cdot g'##?
$$\frac {\partial}{\partial x} \left( x cos(xy) - 2y \right) = (1) cos(xy) + x (-y sin(xy)) = cos(xy) - xy sin(xy)$$
I feel like I'm missing something. Have I done this right? Am I way overthinking this?
 
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  • #2
atomicpedals said:
And this is where I get hung up. Do I simply proceed with an application of the production rule, ##(f \cdot g)' = f' \cdot g + f \cdot g'##?
$$\frac {\partial}{\partial x} \left( x cos(xy) - 2y \right) = (1) cos(xy) + x (-y sin(xy)) = cos(xy) - xy sin(xy)$$
I feel like I'm missing something. Have I done this right? Am I way overthinking this?
That looks right.
 
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  • #3
PeroK said:
That looks right.
Thanks! Sometimes, when you spend too long looking at an exercise, you question your sanity.
 
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To take a partial derivative with respect to x you treat y as a constant. So [itex]\frac{\partial}{\partial x}x cos(xy)- 2x[/itex] is the same as [itex]\frac{d}{dx} xcos(ax)- 2x= cos(ax)- axsin(x)- 2[/itex] where, since I replaced "y" with the constant "a" we need to replace back- [itex]\frac{\partial^2V}{\partial xy}= cos(xy)- xysin(xy)- 2[/itex].

Similarly, [itex]\frac{\partial}{\partial y} y cox(xy)- 2y[/itex] is [itex]\frac{d}{dy} y cos(ay)- 2y= cos(ay)- ay sin(ay)- 2= cos(xy)- xy sin(xy)- 2[/itex].

Notice that those are the same. It is true that, as long as V and its first and second partial derivatives are continuous, [itex]\frac{\partial}{\partial x}\frac{\partial V}{\partial y}= \frac{\partial}{\partial y}\frac{\partial V}{\partial x}[/itex].
 
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Related to Understanding Mixed Partial Derivatives: How Do You Solve Them?

1. What is a partial derivative?

A partial derivative is a mathematical concept used in calculus to measure how a function changes when only one of its variables is changed while keeping all other variables constant.

2. How is a partial derivative different from a regular derivative?

A regular derivative measures the change in a function with respect to a single variable, while a partial derivative measures the change in a function with respect to one variable while holding all other variables constant.

3. What is the notation used for partial derivatives?

The notation used for partial derivatives is similar to regular derivatives, but with a subscript denoting which variable is being held constant. For example, the partial derivative of a function f(x,y) with respect to x would be written as ∂f/∂x.

4. What is the purpose of calculating partial derivatives?

Partial derivatives are used in many fields of science and engineering, such as physics, economics, and engineering, to model and analyze complex systems with multiple variables. They are also used in optimization problems to find the maximum or minimum values of a function.

5. How do you calculate a partial derivative?

To calculate a partial derivative, you first take the derivative of the function with respect to the variable in question, treating all other variables as constants. Then, you evaluate the resulting derivative at the specific value of the variable. This process is repeated for each variable in the function.

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