LeSSOn 223.8 Implicit differentiation: Problem 10 (1 point) \( 9 x^{9} y=4 . \quad x \) and \( y \) are functions of \( t \). Use implicit differentiation to express \( d y / d t \) in terms of \( d x / d t, x \) and \( y \) \( \frac{d y}{d t}=\square \) \( \begin{array}{l}\text { Preview My Answers } \\ \text { You have attempted this problem } 0 \text { times. } \\ \text { You have unlimited attempts remaining. }\end{array} \)

To solve for \( \frac{d y}{d t} \) using implicit differentiation, we start by differentiating both sides of the equation \( 9 x^{9} y = 4 \) with respect to \( t \): \[ \frac{d}{dt}(9 x^{9} y) = \frac{d}{dt}(4). \] Using the product rule on the left side gives: \[ 9 \left( \frac{d}{dt}(x^{9}) y + x^{9} \frac{d y}{d t}\right) = 0. \] Now, differentiating \( x^{9} \) gives us \( 9 x^{8} \frac{d x}{d t} \), so we have: \[ 9 \left( 9 x^{8} \frac{d x}{d t} y + x^{9} \frac{d y}{d t}\right) = 0. \] Dividing everything by 9, we simplify to: \[ (9 x^{8} y) \frac{d x}{d t} + x^{9} \frac{d y}{d t} = 0. \] Now we can solve for \( \frac{d y}{d t} \): \[ x^{9} \frac{d y}{d t} = - (9 x^{8} y) \frac{d x}{d t}, \] which simplifies to: \[ \frac{d y}{d t} = -\frac{9 y}{x} \frac{d x}{d t}. \] So, the final answer is: \[ \frac{d y}{d t} = -\frac{9 y}{x^{8}} \frac{d x}{d t}. \]