Question

15 multiple choice 1 point find dy/dx by implicit differentiation. 8y² - 3x² = 7 options: 3x²/(16y), (6x + 7)/(16y), 3x/8, 3x/(8y)

Explanation:

Step1: Differentiate both sides w.r.t. x

Differentiate \(8y^2 - 3x^2 = 7\) with respect to \(x\). Using the chain rule for \(8y^2\) (since \(y\) is a function of \(x\)) and power rule for \(-3x^2\):
\(\frac{d}{dx}(8y^2) - \frac{d}{dx}(3x^2) = \frac{d}{dx}(7)\)
\(16y \cdot \frac{dy}{dx} - 6x = 0\) (because derivative of constant 7 is 0, and \(\frac{d}{dx}(y^2)=2y\frac{dy}{dx}\), so \(8 \cdot 2y \frac{dy}{dx}=16y\frac{dy}{dx}\); \(\frac{d}{dx}(x^2)=2x\), so \(-3 \cdot 2x = -6x\))

Step2: Solve for \(\frac{dy}{dx}\)

Rearrange the equation to isolate \(\frac{dy}{dx}\):
\(16y \cdot \frac{dy}{dx} = 6x\)
Divide both sides by \(16y\) (assuming \(y
eq 0\)):
\(\frac{dy}{dx} = \frac{6x}{16y}\)
Simplify the fraction by dividing numerator and denominator by 2:
\(\frac{dy}{dx} = \frac{3x}{8y}\)

Answer:

\(\frac{3x}{8y}\) (corresponding to the option with this expression)