Questions: Find dy/dx by implicit differentiation. 20 x^2 = 20 - e^y dy/dx = (Type an exact answer.)

Transcript text: Assignments: FA24 Quiz 2 - Chapter 12 rson.com/Student/PlayerTest.asp... This quiz: 100 point(s) possible Question 13 of 13 This question: 5 point(s) possible Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ by implicit differentiation. \[ \begin{array}{l} 20 x^{2}=20-e^{y} \\ \frac{d y}{d x}=\square \end{array} \] $\square$ (Type an exact answer.) (1) Time Remaining: 00:13:33

Solution

Solution Steps

To find $\frac{dy}{dx}$ using implicit differentiation for the equation $20x^2 = 20 - e^y$, we will differentiate both sides of the equation with respect to $x$. For the left side, apply the power rule. For the right side, apply the chain rule to differentiate $e^y$ with respect to $x$, which involves multiplying by $\frac{dy}{dx}$. Then, solve for $\frac{dy}{dx}$.

Step 1: Differentiate Both Sides

We start with the equation: \[ 20x^2 = 20 - e^y \] Differentiating both sides with respect to $x$, we have: \[ \frac{d}{dx}(20x^2) = \frac{d}{dx}(20 - e^y) \] The left side becomes: \[ \frac{d}{dx}(20x^2) = 40x \] The right side, using the chain rule, becomes: \[ \frac{d}{dx}(20 - e^y) = 0 - e^y \frac{dy}{dx} \]

Step 2: Set Up the Equation

Setting the derivatives equal gives us: \[ 40x = -e^y \frac{dy}{dx} \]

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

Rearranging the equation to solve for $\frac{dy}{dx}$: \[ \frac{dy}{dx} = -\frac{40x}{e^y} \]