question 7 of 15, step 1 of 1 given the equation xy - y^9 = 4y, find dx/dy by implicit differentiation. answer

Explanation:

Step1: Differentiate both sides

Differentiate $xy - y^{9}=4y$ with respect to $y$. Using the product - rule $(uv)^\prime = u^\prime v+uv^\prime$ for the $xy$ term (where $u = x$ and $v = y$), we get: $\frac{d(xy)}{dy}-\frac{d(y^{9})}{dy}=\frac{d(4y)}{dy}$. The derivative of $xy$ with respect to $y$ is $x + y\frac{dx}{dy}$, the derivative of $y^{9}$ with respect to $y$ is $9y^{8}$, and the derivative of $4y$ with respect to $y$ is $4$. So, $x + y\frac{dx}{dy}-9y^{8}=4$.

Step2: Isolate $\frac{dx}{dy}$

First, move the non - $\frac{dx}{dy}$ terms to the other side: $y\frac{dx}{dy}=4 + 9y^{8}-x$. Then, divide both sides by $y$ (assuming $y\neq0$) to solve for $\frac{dx}{dy}$: $\frac{dx}{dy}=\frac{4 + 9y^{8}-x}{y}$.

Answer:

$\frac{4 + 9y^{8}-x}{y}$