Questions: Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point. 6x^2 - 7y^3 = -394 ; (-3,4) dy/dx =

Transcript text: Differentiate implicitly to find $\frac{d y}{d x}$. Then find the slope of the curve at the given point. \[ \begin{array}{l} 6 x^{2}-7 y^{3}=-394 ; \quad(-3,4) \\ \frac{d y}{d x}=\square \end{array} \] $\square$

Solution

Solution Steps

To find $\frac{dy}{dx}$ using implicit differentiation, differentiate both sides of the equation with respect to $x$, applying the chain rule to terms involving $y$. Then, solve for $\frac{dy}{dx}$. To find the slope at the given point $(-3, 4)$, substitute $x = -3$ and $y = 4$ into the expression for $\frac{dy}{dx}$.

Step 1: Implicit Differentiation

We start with the equation given by $ 6x^2 - 7y^3 = -394 $. To find $\frac{dy}{dx}$, we differentiate both sides with respect to $x$: \[ \frac{d}{dx}(6x^2) - \frac{d}{dx}(7y^3) = 0 \] This results in: \[ 12x - 21y^2 \frac{dy}{dx} = 0 \]

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

Rearranging the equation gives us: \[ 21y^2 \frac{dy}{dx} = 12x \] Thus, we can express $\frac{dy}{dx}$ as: \[ \frac{dy}{dx} = \frac{12x}{21y^2} = \frac{4x}{7y^2} \]

Step 3: Evaluate the Slope at the Given Point

Next, we substitute the point $(-3, 4)$ into the expression for $\frac{dy}{dx}$: \[ \frac{dy}{dx} \bigg|_{(-3, 4)} = \frac{4(-3)}{7(4^2)} = \frac{-12}{7 \cdot 16} = \frac{-12}{112} = \frac{-3}{28} \]