Questions: Find dy/dx by implicit differentiation. x^(11 / 3) + y^(11 / 3) = 10 dy/dx = □

Solution Steps

To find \(\frac{d y}{d x}\) by implicit differentiation for the given equation \(x^{11/3} + y^{11/3} = 10\):

1. Differentiate both sides of the equation with respect to \(x\).
2. Apply the chain rule to the \(y\) term since \(y\) is a function of \(x\).
3. Solve for \(\frac{d y}{d x}\).

Given the equation: \[ x^{\frac{11}{3}} + y^{\frac{11}{3}} = 10 \]

Differentiate both sides with respect to \(x\): \[ \frac{d}{dx}\left(x^{\frac{11}{3}} + y^{\frac{11}{3}}\right) = \frac{d}{dx}(10) \]

Applying the chain rule to the \(y\) term: \[ \frac{11}{3} x^{\frac{8}{3}} + \frac{11}{3} y^{\frac{8}{3}} \frac{dy}{dx} = 0 \]

Rearrange the equation to solve for \(\frac{dy}{dx}\): \[ \frac{11}{3} y^{\frac{8}{3}} \frac{dy}{dx} = -\frac{11}{3} x^{\frac{8}{3}} \]

Divide both sides by \(\frac{11}{3} y^{\frac{8}{3}}\): \[ \frac{dy}{dx} = -\frac{x^{\frac{8}{3}}}{y^{\frac{8}{3}}} \]

Simplify the expression: \[ \frac{dy}{dx} = -\left(\frac{x}{y}\right)^{\frac{8}{3}} \]

Final Answer