Question

Calculate the derivative using implicit differentiation:

∂w /∂z,x^(5)w+w^7+wz^2+8yz=0

∂w/ ∂z= ?

Answer 1

Explanation:

Given:
`x^5w + w^7 + wz^2 + 8yz = 0`

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

`x^5\frac{\partial{w}}{\partial{z}} + 7w^6\frac{\partial{w}}{\partial{z}} + z^2\frac{\partial{w}}{\partial{z}} + 2z + 8y =0`

Collecting all terms containing
`\frac{\partial{w}}{\partial{z}},` we get

`(x^5 + 7w^6 + z^2)\frac{\partial{w}}{\partial{z}} = -2(z + 4y)`

Then

`\frac{\partial{w}}{\partial{z}} = (-2(z + 4y))/(x^5 + 7w^6 + z^2)`

Answer 2

The partial derivative
`\((\partial w)/(\partial z)\)` for the given function x^5w + w^7 + wz^2 + 8yz = 0 is
`\(-(2wz + 8y)/(5x^4)\)`, obtained through the process of implicit differentiation.

To find the partial derivative
`\((\partial w)/(\partial z)\)` for the given implicit function
`\(x^5w + w^7 + wz^2 + 8yz = 0\)`, we'll use implicit differentiation.

Start by taking the partial derivative of each term with respect to z:

`\[5x^4 (\partial w)/(\partial z) + 2wz + 8y = 0.\]`

Next, isolate
`\((\partial w)/(\partial z)\)`:

`\[5x^4 (\partial w)/(\partial z) = -2wz - 8y.\]`

Now, solve for
`\((\partial w)/(\partial z)\)`:

`\[(\partial w)/(\partial z) = -(2wz + 8y)/(5x^4).\]`

This is the partial derivative of w with respect to z.

In conclusion, the partial derivative expresses the rate of change of w with respect to z in the context of the given implicit equation.