Question

use implicit differentiation to determine \\(\frac{dy}{dx}\\) for the equation \\(x + y^9 = 6\\).

answer

\\(\frac{dy}{dx} =\\)

Explanation:

Differentiate both sides with respect to \(x\)

$$ \frac{d}{dx}(x + y^9) = \frac{d}{dx}(6) $$

$$ 1 + 9y^8 \frac{dy}{dx} = 0 $$

Solve for \(\frac{dy}{dx}\)

$$ 9y^8 \frac{dy}{dx} = -1 $$

$$ \frac{dy}{dx} = -\frac{1}{9y^8} $$

Answer:

Use implicit differentiation to determine \(\frac{dy}{dx}\) for the equation \(x + y^9 = 6\).

\(\frac{dy}{dx} =\) <blank>\(-\frac{1}{9y^8}\)</blank>