# Suppose that z is implicitly defined by the equation: y z^2 + x^2 ln y = tan z. Find partial z /...

## Question:

Suppose that {eq}z {/eq} is implicitly defined by the equation: {eq}\displaystyle y z^2 + x^2 \ln y = \tan z {/eq}. Find {eq}\dfrac {\partial z}{\partial x} {/eq} and {eq}\displaystyle \dfrac {\partial z} {\partial y} {/eq}.

## Implicit Differentiation of the Function {eq}F(x,y,z) {/eq}:

If we have a function {eq}F(x,y,z) {/eq}, we can calculate {eq}\dfrac{\partial z}{\partial x} {/eq} and {eq}\dfrac{\partial z}{\partial y} {/eq} using the following formulas:

{eq}\dfrac{\partial z}{\partial x} = -\dfrac{F_{x}(x,y,z)}{F_{z}(x,y,z)} {/eq} and {eq}\dfrac{\partial z}{\partial y} = -\dfrac{F_{y}(x,y,z)}{F_{z}(x,y,z)}. {/eq}

The above formulas emphasize the need to derive implicitly to calculate the request.

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Let {eq}yz^2 + x^2 \ln y = \tan z. {/eq}

Let's define {eq}F( x, y,z) = yz^2 + x^2 \ln y - \tan z. {/eq}

Step 1. Calculation of \dfrac{\partial...