given x^5 + 8xy - y^6 = y + 7, use implicit differentiation to find dy/dx. dy/dx = □

Explanation:

Step1: Differentiate each term

Differentiate $x^5$ with respect to $x$ gives $5x^4$, for $8xy$ use product - rule $(uv)' = u'v+uv'$ where $u = 8x$ and $v = y$, so $(8xy)'=8y + 8x\frac{dy}{dx}$, differentiate $-y^6$ with respect to $x$ gives $-6y^5\frac{dy}{dx}$, differentiate $y$ with respect to $x$ gives $\frac{dy}{dx}$ and differentiate $7$ with respect to $x$ gives $0$. So we have: $5x^4+8y + 8x\frac{dy}{dx}-6y^5\frac{dy}{dx}=\frac{dy}{dx}+0$

Step2: Isolate $\frac{dy}{dx}$ terms

Move all terms with $\frac{dy}{dx}$ to one side: $8x\frac{dy}{dx}-6y^5\frac{dy}{dx}-\frac{dy}{dx}=-5x^4 - 8y$ Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(8x-6y^5 - 1)=-5x^4 - 8y$

Step3: Solve for $\frac{dy}{dx}$

$\frac{dy}{dx}=\frac{-5x^4 - 8y}{8x-6y^5 - 1}$

Answer:

$\frac{-5x^4 - 8y}{8x-6y^5 - 1}$