Questions: Given the equation 3x + y^2 = 8 find dy/dx dy/dx = 3/(2y) dy/dx = -3/(2y) dy/dx = 2y/3 dy/dx = -2y/3

Transcript text: Given the equation \[ \begin{array}{l} 3 x+y^{2}=8 \\ \text { find } \frac{d y}{d x} \end{array} \] $\frac{d y}{d x}=\frac{3}{2 y}$ $\frac{d y}{d x}=-\frac{3}{2 y}$ $\frac{d y}{d x}=\frac{2 y}{3}$ $\frac{d y}{d x}=-\frac{2 y}{3}$

Solution

Solution Steps

To find $\frac{d y}{d x}$, we need to use implicit differentiation on the given equation $3x + y^2 = 8$. Differentiate both sides with respect to $x$, applying the chain rule to the $y^2$ term. Solve for $\frac{d y}{d x}$.

Step 1: Differentiate the Equation

Given the equation:

\[ 3x + y^2 = 8 \]

Differentiate both sides with respect to $x$:

\[ \frac{d}{dx}(3x + y^2) = \frac{d}{dx}(8) \]

This gives:

\[ 3 + 2y \frac{dy}{dx} = 0 \]

Step 2: Solve for $\frac{dy}{dx}$

Rearrange the equation to solve for $\frac{dy}{dx}$:

\[ 2y \frac{dy}{dx} = -3 \]

\[ \frac{dy}{dx} = -\frac{3}{2y} \]