Given the equation $$ x y-y^{7}=4 y $$, find $$ \frac{d x}{d y} $$ by implicit differentiation

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We start with the equation

$$
xy - y^7 = 4y.
$$

Since $$ x $$ is a function of $$ y $$, we differentiate both sides with respect to $$ y $$.

1. Differentiate the left-hand side:
   - For $$ xy $$, apply the product rule:
     
     $$
     \frac{d}{dy}(xy) = x \cdot \frac{d}{dy}(y) + y \cdot \frac{dx}{dy} = x + y \frac{dx}{dy}.
     $$
   
   - For $$ y^7 $$, differentiate normally:
     
     $$
     \frac{d}{dy}(y^7) = 7y^6.
     $$
     
     Thus, the derivative of $$ -y^7 $$ is:
     
     $$
     -7y^6.
     $$

2. Differentiate the right-hand side:
   
   $$
   \frac{d}{dy}(4y) = 4.
   $$

3. Combine the results to write the differentiated equation:
   
   $$
   x + y \frac{dx}{dy} - 7y^6 = 4.
   $$

4. Solve for $$ \frac{dx}{dy} $$:
   - Isolate $$ y \frac{dx}{dy} $$:

$$
     y \frac{dx}{dy} = 4 - x + 7y^6.
     $$
     
   - Divide both sides by $$ y $$ (assuming $$ y \neq 0 $$):

$$
     \frac{dx}{dy} = \frac{4 - x + 7y^6}{y}.
     $$

Thus, the derivative $$ \frac{dx}{dy} $$ is

$$
\frac{dx}{dy} = \frac{4 - x + 7y^6}{y}.
$$
$$ \frac{dx}{dy} = \frac{4 - x + 7y^6}{y} $$

Given the equation \( x y-y^{7}=4 y \), find \( \frac{d x}{d y} \) by implicit differentiation

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