Find $\frac{d y}{d x}$ using implicit differentiation.
$3 x^{3} = 5 y^{2} + 9 y$

Answer

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Answer

Final Answer: The derivative of $y$ with respect to $x$ is $\frac{9 x^{2}}{10 y + 9}$ .

Steps

Step 1 ：Differentiate both sides of the equation $3 x^{3} = 5 y^{2} + 9 y$ with respect to $x$ .

Step 2 ：The derivative of $3 x^{3}$ with respect to $x$ is $9 x^{2}$ .

Step 3 ：The derivative of $5 y^{2}$ with respect to $x$ is $10 y \frac{d y}{d x}$ .

Step 4 ：The derivative of $9 y$ with respect to $x$ is $9 \frac{d y}{d x}$ .

Step 5 ：After differentiating, we get the equation $9 x^{2} = 10 y \frac{d y}{d x} + 9 \frac{d y}{d x}$ .

Step 6 ：Solve for $\frac{d y}{d x}$ to get $\frac{d y}{d x} = \frac{9 x^{2}}{10 y + 9}$ .

Step 7 ：Final Answer: The derivative of $y$ with respect to $x$ is $\frac{9 x^{2}}{10 y + 9}$ .