Given the equation $$ x-y^{8}=3 $$, find $$ \frac{d x}{d y} $$ by implicit differentiation.

Question

UpStudy AI · Accepted Answer

1. Differentiate both sides of the equation with respect to $$ y $$:

$$
   \frac{d}{dy}(x) - \frac{d}{dy}(y^8) = \frac{d}{dy}(3)
   $$

2. Compute each derivative:

- For $$ x $$, since $$ x $$ is a function of $$ y $$:

$$
     \frac{d}{dy}(x) = \frac{dx}{dy}
     $$

- For $$ y^8 $$:

$$
     \frac{d}{dy}(y^8) = 8y^7
     $$

- For the constant $$ 3 $$:

$$
     \frac{d}{dy}(3) = 0
     $$

3. Substitute the derivatives back into the equation:

$$
   \frac{dx}{dy} - 8y^7 = 0
   $$

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

$$
   \frac{dx}{dy} = 8y^7
   $$
$$ \frac{dx}{dy} = 8y^7 $$

Given the equation \( x-y^{8}=3 \), find \( \frac{d x}{d y} \) by implicit differentiation.