How can we use the equations to determine the partial derivatives dz/dx and dz/dy for the equation x² + 4y² + 3z² = 1?

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To find the partial derivatives
dz/dx and dz/dy of the implicit function given by the equation x² + 4y² + 3z² = 1, we can follow these steps:

Step 1: Differentiate the equation with respect to x

Using implicit differentiation, we differentiate both sides of the equation with respect to x:

 d/dx (x²) + d/dx (4y²) + d/dx (3z²) = d/dx (1)

This gives us:

 2x + 8y(dy/dx) + 6z(dz/dx) = 0

Step 2: Solve for dz/dx

Now, we can rearrange this equation to solve for dz/dx:

 6z(dz/dx) = -2x - 8y(dy/dx)

Dividing both sides by 6z:

 dz/dx = (-2x - 8y(dy/dx)) / (6z)

Step 3: Differentiate the equation with respect to y

Next, we differentiate the original equation with respect to y:

 d/dy (x²) + d/dy (4y²) + d/dy (3z²) = d/dy (1)

This gives us:

 0 + 8y + 6z(dz/dy) = 0

Step 4: Solve for dz/dy

Rearranging the above equation provides us with an expression for dz/dy:

 6z(dz/dy) = -8y

Dividing both sides by 6z:

 dz/dy = -8y / (6z)

Final Summary

In summary:

  • dz/dx: (-2x - 8y(dy/dx)) / (6z)
  • dz/dy: -8y / (6z)

This method allows us to effectively find the partial derivatives of z with respect to x and y while adhering to the constraints of the given equation.

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