Partial Derivatives of xu^2 + yv = 2 at (1,1)

In summary, the equations xu^2 + yv = 2 and 2yv^2 + xu = 3 define u(x,y) and v(x,y) near the point (x,y) = (1,1) and (u,v) = (1,1). The partial derivatives for u and v are: (A) ∂u/∂x(1,1) = -0.428571428571429, (B) ∂u/∂y(1,1) = -0.285714285714286, (C) ∂v/∂x(1,1) = -0.142857142857143, and (D)
  • #1
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Homework Statement


The equations xu^2 + yv = 2, 2yv^2 + xu = 3 define u(x,y) and v(x,y) in terms of x and y near the point (x,y) = (1,1) and (u,v) = (1,1).

Compute the following partial derivatives:
(A) ∂u/∂x(1,1)
(B) ∂u/∂y(1,1)
(C) ∂v/∂x(1,1)
(D) ∂v/∂y(1,1)

The answers are:
(A) ∂u/∂x(1,1) = -0.428571428571429
(B) ∂u/∂y(1,1) = -0.285714285714286
(C) ∂v/∂x(1,1) = -0.142857142857143
(D) ∂v/∂y(1,1) = -0.428571428571429

Homework Equations


To my knowledge: partial differentiation and implicit differentiation.

The Attempt at a Solution


I tried implicitly and partially differentiating xu^2 + yv = 2 and got:
u^2 + 2xu∂u/∂x = 0
∂u/∂x = -u^2 /(2xu)
∂u/∂x(1,1) = -(1)^2/(2*1*1) = -1/2 (which is close to the answer but not good enough).

Could someone please tell me what I am doing wrong and how to do this correctly?

Any help would be greatly appreciated!
Thanks in advance!
 
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  • #2


Where is the ∂v/∂x term in your partial derivative? You'll need to differentiate both equations with respect to both x and y and then treat it as a system of linear equations in the four unknowns.
 
  • #3


Your advice worked for me. Thanks!
 

1. What is the formula for calculating partial derivatives?

The formula for calculating partial derivatives is ∂f/∂x, where f is the function and x is the variable with respect to which the derivative is being calculated.

2. How do you calculate partial derivatives of a function with multiple variables?

To calculate partial derivatives for a function with multiple variables, treat all other variables as constants and differentiate the function with respect to the variable of interest.

3. What is the purpose of calculating partial derivatives?

Calculating partial derivatives allows us to determine the rate of change of a function with respect to a specific variable, while holding all other variables constant.

4. What is the significance of partial derivatives in scientific research?

Partial derivatives are commonly used in scientific research to analyze and model complex systems with multiple variables, such as in physics, engineering, and economics.

5. How do you interpret the result of a partial derivative calculation?

The result of a partial derivative calculation represents the slope of the tangent line to the surface of the function at a specific point, and can be used to determine the direction and magnitude of change in the function with respect to the variable of interest.

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