Question

Find \(d y / d x\) by implicit differentiation. $$ x^{3}+y^{3}=8 $$

Short answer

The derivative \( \frac {dy}{dx}\) of the function given by \(x^{3} + y^{3} = 8\) is \(- \frac{x^{2}}{y^{2}}\).

Step-by-step solution

Differentiate Both Sides of the Equation

Differentiate both sides of the equation with respect to \(x\), using the power rule. The derivative of \(x^{3}\) is \(3x^{2}\). Then differentiate \(y^{3}\) using the chain rule, which yields \(3y^{2}\frac {dy}{dx}\). The derivative of a constant \(8\) is \(0\) because a constant does not change. So, we get:\[3x^{2} + 3y^{2}\frac {dy}{dx} = 0.\]

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

Rearrange the equation to solve for \(\frac {dy}{dx}\). To isolate \(\frac {dy}{dx}\), subtract \(3x^{2}\) from both sides and divide by \(3y^{2}\):\[\frac {dy}{dx} = - \frac {3x^{2}} {3y^{2}}. \]

Simplify the Expression

Simplify the expression for \(\frac {dy}{dx}\) by cancelling out the common factor 3:\[\frac {dy}{dx} = - \frac {x^{2}} {y^{2}}.\]