Partial Differentiation of this Equation in x and y

In summary, the partial differentiation of a given equation with respect to the variables x and y involves computing the rate of change of the function while holding one variable constant at a time. This process allows for the analysis of how the function behaves in relation to each variable independently, facilitating the understanding of multi-variable systems. The results yield partial derivatives, which can be used to identify critical points, optimize functions, and explore the relationships between variables in various applications.
  • #1
Martyn Arthur
118
20
Homework Statement
Trying to get to fxx
Relevant Equations
Please see screen print
Hi;
please see below I am trying to understand how to get to the 2 final functions. They should be the same but 6 for the first one and 2 for the second?
(I hope my writing is more clear than previously)
There is an additional question below.
thanks
martyn
1707919506461.png

I can't find a standard derivative for cos^2 theta?
 
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  • #2
Martyn Arthur said:
I can't find a standard derivative for cos^2 theta?

Use the chain rule.
 
  • #3
Please show your work and don’t simply post images of your result. Type out your work.
 
  • #4
Your two first partials are correct, but your notation isn't.
These aren't f(x) and f(y) as you wrote. They are ##f_x(x, y)## and ##f_y(x,y)## respectively. They can also be written more compactly as ##f_x## and ##f_y##.
Martyn Arthur said:
I can't find a standard derivative for cos^2 theta?
It might be helpful to think of this as ##(\cos(\theta))^2## and then use the chain rule, as @pasmith recommended.

Orodruin said:
Please show your work and don’t simply post images of your result. Type out your work.
I agree. In the lower left corner, click on the link that says "LaTeX Guide." A few minutes spent reading that will be very helpful.
 
  • #5
Martyn Arthur said:
Homework Statement: Trying to get to fxx
Relevant Equations: Please see screen print

Hi;
please see below I am trying to understand how to get to the 2 final functions. They should be the same but 6 for the first one and 2 for the second?
No. Why are you saying that?

If you can solve that ##f_x(x,y) = 6x-2y-10## then I'm sure that you can calculate ##f_{x,y}(x,y)##. It's simply the derivative of ##6x-2y-10## with respect to ##y##.
Do something similar for ##f_{y,x}##.
 

FAQ: Partial Differentiation of this Equation in x and y

What is partial differentiation?

Partial differentiation is a technique used in calculus to differentiate a function of multiple variables with respect to one variable while keeping the other variables constant. It is used to find the rate of change of the function with respect to one of its variables.

How do you perform partial differentiation with respect to x?

To perform partial differentiation of a function with respect to x, you treat y (and any other variables) as constants and differentiate the function with respect to x as you would in single-variable calculus. For example, if you have a function f(x, y), the partial derivative with respect to x is denoted as ∂f/∂x.

How do you perform partial differentiation with respect to y?

To perform partial differentiation of a function with respect to y, you treat x (and any other variables) as constants and differentiate the function with respect to y. For example, if you have a function f(x, y), the partial derivative with respect to y is denoted as ∂f/∂y.

What is the significance of second-order partial derivatives?

Second-order partial derivatives provide information about the curvature of the function's surface. They are obtained by differentiating the first-order partial derivatives. For example, the second-order partial derivative with respect to x is denoted as ∂²f/∂x². Mixed partial derivatives, such as ∂²f/∂x∂y, indicate how the function changes as both variables vary simultaneously.

Can partial derivatives be used to find local extrema of a function?

Yes, partial derivatives can be used to find local extrema (maxima, minima, and saddle points) of a function of multiple variables. By setting the first-order partial derivatives equal to zero, you can find the critical points. Then, the second-order partial derivatives can be used in the second-derivative test to classify these critical points.

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