Questions: Find the derivative of y with respect to x by implicit differentiation. dy/dx = x^2 - 5 ln(y) + y^2 = 19

Transcript text: Assignment Submission \& Scoring Assignment Submission For this assignment, you submit answers by questio Assignment Scoring Your best submission for each question part is used 7. [-/1 Points] DETAILS MY NOTES Find $\frac{d y}{d x}$ by implicit differentiation. \[ \frac{d y}{d x}=x^{2}-5 \ln (y)+y^{2}=19 \] Submit Answer

Solution

Solution Steps

To find $\frac{d y}{d x}$ using implicit differentiation, we differentiate both sides of the given equation with respect to $x$. For terms involving $y$, apply the chain rule, treating $y$ as a function of $x$. After differentiating, solve for $\frac{d y}{d x}$.

Step 1: Differentiate the Equation

Given the equation: \[ x^2 + y^2 - 5\ln(y) = 19 \] we differentiate both sides with respect to $x$. The left side becomes: \[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) - 5\frac{d}{dx}(\ln(y)) \] which simplifies to: \[ 2x + 2y\frac{dy}{dx} - \frac{5}{y}\frac{dy}{dx} \]

Step 2: Set the Derivative Equal to Zero

Since the right side of the original equation is a constant (19), its derivative is zero. Therefore, we set the differentiated left side equal to zero: \[ 2x + 2y\frac{dy}{dx} - \frac{5}{y}\frac{dy}{dx} = 0 \]

Step 3: Solve for $\frac{dy}{dx}$

Rearrange the equation to solve for $\frac{dy}{dx}$: \[ 2y\frac{dy}{dx} - \frac{5}{y}\frac{dy}{dx} = -2x \] Factor out $\frac{dy}{dx}$: \[ \frac{dy}{dx}(2y - \frac{5}{y}) = -2x \] Solve for $\frac{dy}{dx}$: \[ \frac{dy}{dx} = \frac{-2x}{2y - \frac{5}{y}} \]