use implicit differentiation to find $\\frac{dy}{dx}$.\n$(6xy + 7)^2 = 24y$\n$\\frac{dy}{dx}=\\square$
use implicit differentiation to find $\\frac{dy}{dx}$.\n$(6xy + 7)^2 = 24y$\n$\\frac{dy}{dx}=\\square$
Answer
Explanation:
Step1: Differentiate both sides with respect to (x)
Use the chain rule ((u^2)^\prime = 2u\cdot u^\prime) for the left - hand side ((u = 6xy + 7)) and ((24y)^\prime=24y^\prime) for the right - hand side. For the left - hand side: [ \begin{align*} \frac{d}{dx}(6xy + 7)^2&=2(6xy + 7)\cdot\frac{d}{dx}(6xy + 7)\ &=2(6xy + 7)\left(6y+6x\frac{dy}{dx}\right) \end{align*} ] For the right - hand side: (\frac{d}{dx}(24y)=24\frac{dy}{dx})
Step2: Expand and collect terms with (\frac{dy}{dx})
Expand the left - hand side: (2(6xy + 7)(6y+6x\frac{dy}{dx})=12(6xy + 7)y+12x(6xy + 7)\frac{dy}{dx}) So the equation becomes (12(6xy + 7)y+12x(6xy + 7)\frac{dy}{dx}=24\frac{dy}{dx}) Move the terms with (\frac{dy}{dx}) to one side: [ \begin{align*} 12x(6xy + 7)\frac{dy}{dx}-24\frac{dy}{dx}&=- 12(6xy + 7)y\ \frac{dy}{dx}\left[12x(6xy + 7)-24\right]&=-12(6xy + 7)y \end{align*} ]
Step3: Solve for (\frac{dy}{dx})
[ \begin{align*} \frac{dy}{dx}&=\frac{-12(6xy + 7)y}{12x(6xy + 7)-24}\ &=\frac{- (6xy + 7)y}{x(6xy + 7)-2}\ &=\frac{-6xy^2-7y}{6x^{2}y+7x - 2} \end{align*} ]
Answer:
(\frac{-6xy^{2}-7y}{6x^{2}y + 7x-2})