Questions: cot(y) = 3x - 8y dy/dx = □

Transcript text: \[ \begin{array}{r} \cot (y)=3 x-8 y \\ \frac{d y}{d x}=\square \end{array} \]

Solution

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.

Step 1: Differentiate the Equation

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

Step 2: Apply Implicit Differentiation

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 \(3 - 8\frac{dy}{dx}\).

Step 3: Solve for \(\frac{dy}{dx}\)

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: \[ \frac{dy}{dx} = -3 \]