) Suppose that z=f(x,y)z=f(x,y) is defined implicitly by an equation of the form F(x,y,z)=0F(x,y,z)=0. Find formulas for the partial derivatives ∂f∂x∂f∂x and ∂f∂y∂f∂y in terms of F1,F2,F3F1,F2,F3.
To enter your answer use F1, F2, F3 as the partial derviatives of FF with respect to its first, second, and third variables.
) Suppose that z=f(x,y)z=f(x,y) is defined implicitly by an equation of the form F(x,y,z)=0F(x,y,z)=0. Find formulas for the partial derivatives ∂f∂x∂f∂x and ∂f∂y∂f∂y in terms of F1,F2,F3F1,F2,F3.
To enter your answer use F1, F2, F3 as the partial derviatives of FF with respect to its first, second, and third variables.
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"z=f(x, y) is defined implicitly by an equation of form F(x, y, z)=0"
The second result, "Implicit Differentiation..." (a pdf) will be helpful.
To find the partial derivatives ∂f/∂x and ∂f/∂y in terms of F1, F2, and F3, we can use the chain rule.
For ∂f/∂x:
∂F/∂x + ∂F/∂z * ∂z/∂x = 0
Since z = f(x,y),
∂F/∂x + ∂F/∂z * ∂f/∂x = 0
Rearranging the equation, we can solve for ∂f/∂x:
∂f/∂x = - (∂F/∂x) / (∂F/∂z)
So, the formula for ∂f/∂x in terms of F1, F2, and F3 is:
∂f/∂x = - (F1) / (F3)
Similarly, for ∂f/∂y:
∂F/∂y + ∂F/∂z * ∂z/∂y = 0
Since z = f(x,y),
∂F/∂y + ∂F/∂z * ∂f/∂y = 0
Rearranging the equation, we can solve for ∂f/∂y:
∂f/∂y = - (∂F/∂y) / (∂F/∂z)
So, the formula for ∂f/∂y in terms of F1, F2, and F3 is:
∂f/∂y = - (F2) / (F3)
To find the partial derivatives ∂f/∂x and ∂f/∂y in terms of F1, F2, and F3, we can use the implicit function theorem.
The implicit function theorem states that if F(x, y, z) = 0 defines z implicitly as a function of x and y in some neighborhood of a point (a, b, c), and if F is continuously differentiable, then the partial derivatives of z with respect to x and y can be expressed in terms of the partial derivatives of F with respect to x, y, and z.
Let's apply this theorem to find the partial derivatives.
1. First, we differentiate both sides of the equation F(x, y, z) = 0 with respect to x:
∂F/∂x + ∂F/∂z * ∂z/∂x = 0
2. Solving for ∂z/∂x, we get:
∂z/∂x = - (∂F/∂x) / (∂F/∂z)
3. Next, we differentiate both sides of the equation F(x, y, z) = 0 with respect to y:
∂F/∂y + ∂F/∂z * ∂z/∂y = 0
4. Solving for ∂z/∂y, we get:
∂z/∂y = - (∂F/∂y) / (∂F/∂z)
So, the formulas for the partial derivatives ∂f/∂x and ∂f/∂y in terms of F1, F2, F3 would be:
∂f/∂x = - (F1) / (F3)
∂f/∂y = - (F2) / (F3)
These formulas express the partial derivatives of the implicitly defined function f in terms of the partial derivatives of the equation F with respect to its variables.