Suppose that a function z = f(x, y) is defined implicitly by constraint (x^2 + y^2 + z^2)^2 = x -...


Suppose that a function {eq}z=f(x,y) {/eq} is defined implicitly by constraint {eq}(x^{2}+y^{2}+z^{2})^{2}=x-y+z {/eq}. Use implicit differentiation to calculate

{eq}\frac{\partial z}{\partial x} {/eq} and {eq}\frac{\partial z}{\partial y} {/eq} in terms of {eq}x,y,z {/eq}.

Implicit Derivative:

We can also calculate its derivative to implicitly defined functions.

For this we can support the resource of partial derivatives that allows us to use the following general result:

{eq}\large F\left( {x,y,z} \right) = 0 \to \left\{ \begin{array}{l} \frac{{\partial z}}{{\partial x}} = - \frac{{\frac{{\partial F}}{{\partial x}}}}{{\frac{{\partial F}}{{\partial z}}}}\\\large \frac{{\partial z}}{{\partial y}} = - \frac{{\frac{{\partial F}}{{\partial y}}}}{{\frac{{\partial F}}{{\partial z}}}} \end{array} \right. {/eq}

Answer and Explanation: 1

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Rewriting the equation in terms of a function:

{eq}{({x^2} + {y^2} + {z^2})^2} = x - y + z\\ F\left( {x,y,z} \right) = {({x^2} + {y^2} + {z^2})^2} -...

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Implicit Functions


Chapter 1 / Lesson 11

Implicit functions in math are equations that depend on both x and y, neither of which can be separated. Learn more about these functions in relation to ovals and circles, and review an example of an implicit function.

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