Partial Derivatives: Calculating fₓ and fᵧ (3,1)

In summary, The conversation is about finding the partial derivatives of a function f(x,y) = g(x-2y) where g(x) is a function of one variable. The question asks for the values of f subscript x and f subscript y at the point (3,1) given that g'(1) = 3. The conversation also mentions using the chain rule for this setup and finding the partial derivatives of f using the definition of a partial derivative.
  • #1
Jessica21
5
0
Hi everyone!

I was wondering if someone could help me with the following question with partial derivatives.

A function f: R^2 -> R is defined by f(x,y) = g(x-2y), where g: R-> R.
If g'(1)= 3, calculate f subscript x (3,1) and f subscript y of (3,1).

thanks!
 
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  • #2
You can use the little X2 icon to type fx and fy.

Do you know the chain rule for this setup:

g is a function of u and u is a function of x and y, then

gx = ? and gy = ?

That's what you need.
 
  • #3
hi jessica21, welcome to pf ;) usually it helps if you show some working...

i would start by trying to find the partial derviatives of f & thinking about what a partial derviative means

[tex] f_x = \frac{\partial f}{\partial x} = \frac{\partial }{\partial x} g(x-2y) [/tex]
 
  • #4
also remember a partial derivative means the other variables are kept constant in the differentiation
 

1. What are partial derivatives?

Partial derivatives are a type of derivative that measures the rate of change of a multivariable function with respect to one of its variables, while holding all other variables constant. It is denoted by ∂f/∂x or fₓ, where x is the independent variable.

2. How do I calculate partial derivatives?

To calculate a partial derivative, you first need to determine which variable you are differentiating with respect to. Then, treat all other variables as constants and use the standard rules of differentiation to find the derivative. For example, if you have the function f(x,y) = x²y + 3xy, and you want to find fₓ at the point (3,1), you would differentiate with respect to x and plug in the values of x=3 and y=1.

3. What is the significance of calculating fₓ and fᵧ at a specific point?

The partial derivatives fₓ and fᵧ at a specific point (a,b) represent the slope of the tangent line in the x and y directions, respectively. This allows us to better understand the behavior of a function in a specific direction and can be useful for optimization problems in multivariable calculus.

4. How do I interpret the values of fₓ and fᵧ?

The values of fₓ and fᵧ represent the rate of change of the function in the x and y directions, respectively. A positive value indicates an increasing function, while a negative value indicates a decreasing function. The magnitude of the value also represents the steepness of the function in that direction.

5. Can I use partial derivatives to find the maximum or minimum of a function?

Yes, partial derivatives can be used to find the maximum or minimum of a function. When finding the critical points of a multivariable function, we set both fₓ and fᵧ equal to 0 and solve for the variables. Then, we use the second derivative test to determine if the critical point is a maximum, minimum, or saddle point.

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