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Step-by-step explanation:

Given: [tex]x^5w + w^7 + wz^2 + 8yz = 0[/tex]

Taking the partial derivative of the above equation with respect to z, we get

[tex]x^5\dfrac{\partial{w}}{\partial{z}} + 7w^6\dfrac{\partial{w}}{\partial{z}} + z^2\dfrac{\partial{w}}{\partial{z}} + 2z + 8y =0[/tex]

Collecting all terms containing [tex]\frac{\partial{w}}{\partial{z}},[/tex] we get

[tex](x^5 + 7w^6 + z^2)\dfrac{\partial{w}}{\partial{z}} = -2(z + 4y)[/tex]

Then

[tex]\dfrac{\partial{w}}{\partial{z}} = \dfrac{-2(z + 4y)}{x^5 + 7w^6 + z^2}[/tex]

The derivative using implicit differentiation can be calculated and after rearranging the terms.

What is implicit differentiation?

Implicit differentiation is the technique in which the derivative of y with respect to x using implicit differentiation without having to solve the supplied equation for y.

We have:

[tex]\rm x^5w+w^7+wz^2+8yz=0[/tex]

We have to find the:

= ∂w/ ∂z

[tex]\rm \dfrac{\partial }{\partial z}[\rm x^5w+w^7+wz^2+8yz=0][/tex]

[tex]\rm x^5\dfrac{\partial w}{\partial z} +7w^6\dfrac{\partial w}{\partial z}+z^2\dfrac{\partial w}{\partial z}+2z+8y=0[/tex]

After rearranging the terms:

[tex]\rm (x^5 +7w^6+z^2)\dfrac{\partial w}{\partial z}= -2(z+4y)[/tex]

[tex]\rm \dfrac{\partial w}{\partial z}= \dfrac{-2(z+4y)}{x^5 +7w^6+z^2}[/tex]

Thus, the derivative using implicit differentiation can be calculated and after rearranging the terms.

Learn more about the implicit differentiation here:

https://brainly.com/question/2292940

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