Find $$ \frac{d y}{d x} $$ by implicit differentiation. $$ x^{8}+y^{5}=4 $$ $$ \frac{d y}{d x}=\square $$

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1. Differentiate both sides of the equation with respect to $$ x $$:
   $$
   \frac{d}{dx}\left(x^{8}\right) + \frac{d}{dx}\left(y^{5}\right) = \frac{d}{dx}(4)
   $$

2. Compute the derivatives:
   - The derivative of $$ x^{8} $$ is:
     $$
     8x^{7}
     $$
   - The derivative of $$ y^{5} $$ with respect to $$ x $$ requires the chain rule:
     $$
     5y^{4}\frac{dy}{dx}
     $$
   - The derivative of the constant $$ 4 $$ is:
     $$
     0
     $$

3. Substitute these into the equation:
   $$
   8x^{7} + 5y^{4}\frac{dy}{dx} = 0
   $$

4. Solve for $$ \frac{dy}{dx} $$:
   $$
   5y^{4}\frac{dy}{dx} = -8x^{7}
   $$
   $$
   \frac{dy}{dx} = -\frac{8x^{7}}{5y^{4}}
   $$

Thus, the derivative is:
$$
\frac{dy}{dx}=-\frac{8x^{7}}{5y^{4}}
$$
$$ \frac{dy}{dx} = -\frac{8x^{7}}{5y^{4}} $$

Find by implicit differentiation.

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