Calculate the derivative using implicit differentiation:
$$
\begin{array}{r}
\frac{\partial w}{\partial z}, \quad x^{9} w+w^{5}+w z^{2}+5 y z=0 \
\frac{\partial w}{\partial z}=\square
\end{array}
$$

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To differentiate the equation $ x^9 w + w^5 + w z^2 + 5 y z = 0 $ implicitly with respect to $ z $, we apply the product rule and chain rule: ‖ ‖ step 2 ⋮ Differentiating each term gives: $ 9x^8 w + x^9 \frac{\partial w}{\partial z} + z^2 \frac{\partial w}{\partial z} + 2wz \frac{\partial z}{\partial z} + 5y \frac{\partial z}{\partial z} + 5z \frac{\partial y}{\partial z} = 0 $ ‖ ‖ step 3 ⋮ Rearranging the equation, we have: $ x^9 \frac{\partial w}{\partial z} + z^2 \frac{\partial w}{\partial z} = - (9x^8 w + 2wz + 5y + 5z \frac{\partial y}{\partial z}) $ ‖ ‖ step 4 ⋮ Factoring out $ \frac{\partial w}{\partial z} $: $ \frac{\partial w}{\partial z} (x^9 + z^2) = - (9x^8 w + 2wz + 5y + 5z \frac{\partial y}{\partial z}) $ ‖ ‖ step 5 ⋮ Finally, solving for $ \frac{\partial w}{\partial z} $: $ \frac{\partial w}{\partial z} = -\frac{9x^8 w + 2wz + 5y + 5z \frac{\partial y}{\partial z}}{x^9 + z^2} $ ‖ ∻Answer∻ ⚹ $ \frac{\partial w}{\partial z} = -\frac{9x^8 w + 2wz + 5y + 5z \frac{\partial y}{\partial z}}{x^9 + z^2} $ ⚹ ∻Key Concept∻ ⚹Implicit differentiation allows us to find derivatives of variables that are defined implicitly rather than explicitly.⚹ ∻Explanation∻ ⚹The answer shows how to differentiate an equation involving multiple variables, leading to the expression for $ \frac{\partial w}{\partial z} $ in terms of the other variables.⚹◊What is the implicit differentiation formula used to find $\frac{\partial w}{\partial z}$ in the equation $x^{9} w + w^{5} + wz^{2} + 5yz = 0$?⍭ Generate me a similar question◊

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