Suppose that z=f(x,y) is defined implicitly by an equation of the form F(x,y,z)=0. Find formulas for the partial derivatives ∂f/∂x and ∂f/∂y in terms of F1,F2,F3
To enter your answer use F1, F2, F3 as the partial derviatives of Fwith respect to its first, second, and third variables.
To find the formulas for 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 we have an equation of the form F(x, y, z) = 0 and we can solve it for z as a function of x and y, then we can find the partial derivatives of z with respect to x and y.
Let's start by differentiating both sides of the equation F(x, y, z) = 0 with respect to x:
∂F/∂x + ∂F/∂z * ∂z/∂x = 0
Now, let's solve for ∂z/∂x:
∂z/∂x = - (∂F/∂x) / (∂F/∂z)
Using the chain rule, we can express ∂F/∂x and ∂F/∂z in terms of the partial derivatives of F, F1, F2, and F3:
∂F/∂x = ∂F1/∂x + ∂F2/∂x + ∂F3/∂x * ∂z/∂x
∂F/∂z = ∂F1/∂z + ∂F2/∂z + ∂F3/∂z * ∂z/∂x
Substituting these expressions back into our equation for ∂z/∂x, we get:
∂z/∂x = - ( ∂F1/∂x + ∂F2/∂x + ∂F3/∂x * ∂z/∂x ) / ( ∂F1/∂z + ∂F2/∂z + ∂F3/∂z * ∂z/∂x )
Simplifying the equation by multiplying both sides by ( ∂F1/∂z + ∂F2/∂z + ∂F3/∂z * ∂z/∂x ), we get:
( ∂F1/∂z + ∂F2/∂z + ∂F3/∂z * ∂z/∂x ) * ∂z/∂x = - ( ∂F1/∂x + ∂F2/∂x + ∂F3/∂x * ∂z/∂x )
Now, solving for ∂z/∂x, we have:
∂z/∂x = - ( ∂F1/∂x + ∂F2/∂x + ∂F3/∂x * ∂z/∂x ) / ( ∂F1/∂z + ∂F2/∂z + ∂F3/∂z * ∂z/∂x )
This equation relates the partial derivative ∂z/∂x to the partial derivatives F1, F2, and F3.
Similarly, we can find the formula for the partial derivative ∂z/∂y using the same steps, but differentiating with respect to y instead of x:
∂z/∂y = - ( ∂F1/∂y + ∂F2/∂y + ∂F3/∂y * ∂z/∂x ) / ( ∂F1/∂z + ∂F2/∂z + ∂F3/∂z * ∂z/∂x )
These formulas express the partial derivatives ∂z/∂x and ∂z/∂y in terms of the partial derivatives F1, F2, and F3.