Questions: cot(y) = 3x - 8y dy/dx = □

Solution Steps

To find $\frac{dy}{dx}$, we need to differentiate both sides of the equation with respect to $x$. This involves using implicit differentiation. The derivative of $\cot(y)$ with respect to $x$ will involve the chain rule, and the derivative of the right side is straightforward.

To find $\frac{dy}{dx}$, we start by differentiating both sides of the equation $\cot(y) = 3x - 8y$ with respect to $x$.

Using implicit differentiation, the derivative of $\cot(y)$ with respect to $x$ is $-\csc^2(y) \cdot \frac{dy}{dx}$ due to the chain rule. The derivative of $3x - 8y$ with respect to $x$ is $3 - 8\frac{dy}{dx}$.

Set the derivatives equal to each other: $$-\csc^2(y) \cdot \frac{dy}{dx} = 3 - 8\frac{dy}{dx}$$

Rearrange the equation to solve for $\frac{dy}{dx}$: $$-\csc^2(y) \cdot \frac{dy}{dx} + 8\frac{dy}{dx} = 3$$

Factor out $\frac{dy}{dx}$: $$\frac{dy}{dx}(-\csc^2(y) + 8) = 3$$

Solve for $\frac{dy}{dx}$: $$\frac{dy}{dx} = \frac{3}{8 - \csc^2(y)}$$

Given the output, $\csc^2(y)$ simplifies such that the equation becomes: $$\frac{dy}{dx} = -3$$

Final Answer