Partial derivatives - verify solution?

In summary, the conversation discusses finding the partial derivatives of a given function F in terms of the partial derivatives of f and g. It also addresses the issue of finding the partial derivatives of g in the case that F is not a function of g.
  • #1
jjou
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[SOLVED] partial derivatives - verify solution?

Let [tex]f:\mathbb{R}^3\rightarrow\mathbb{R}[/tex], [tex]g:\mathbb{R}^2\rightarrow\mathbb{R}[/tex], and [tex]F:\mathbb{R}^2\rightarrow\mathbb{R}[/tex] be given by
[tex]F(x,y)=f(x,y,g(x,y))[/tex].
1. Find DF in terms of the partial derivatives of f and g.
2. If F(x,y)=0 for all (x,y), find [tex]D_1g[/tex] and [tex]D_2g[/tex] in terms of the partial derivatives of f.

My solution:
1. [tex]DF=D_1F+D_2F=(f_1+f_3g_1)+(f_2+f_3g_2)[/tex]
2. If [tex]f_3\neq0[/tex], then we have the partials of F being zero, so:
[tex]g_1=-f_1/f_3[/tex] and [tex]g_2=-f_2/f_3[/tex]. However, if [tex]f_3=0[/tex] then we have [tex]f_1=f_2=0[/tex].

My concern is with the last part of 2. If [tex]f_3=0[/tex], then I cannot make any statement about the partials of g. Am I doing something wrong?


NOTE: [tex]f_1[/tex] refers to differentiation of f by the first variable.
 
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  • #2
No, that's completely true. If F is, in fact, NOT a function of g, then no information about F can tell you anything about g!
 
  • #3
Thanks! :)
 

Related to Partial derivatives - verify solution?

1. What is a partial derivative?

A partial derivative is a mathematical concept that measures the instantaneous rate of change of a function with respect to one of its variables, while holding all other variables constant.

2. Why do we use partial derivatives?

Partial derivatives are used to determine the sensitivity of a function to changes in its variables. They are especially useful in multivariable calculus and optimization problems.

3. How do you verify a solution using partial derivatives?

To verify a solution using partial derivatives, you need to calculate the partial derivatives of the given function and substitute the values of the variables in the solution into these partial derivatives. If the resulting values are equal, then the solution is verified.

4. What are some common applications of partial derivatives?

Partial derivatives are commonly used in physics, economics, engineering, and other fields 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. Can you provide an example of how to verify a solution using partial derivatives?

Sure, let's say we have the function f(x,y) = 3x^2 + 2y + 1 and the solution (x,y) = (2,3). To verify this solution, we calculate the partial derivatives of f with respect to x and y, which are 6x and 2 respectively. Substituting x = 2 and y = 3 into these partial derivatives, we get 12 and 2. Since these values match the given solution, we can conclude that (2,3) is a valid solution to the function.

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