If x^3+y^3=6xy, what is (dy)/(dx)?

Exercise · ZW-IMPLICIT-DIFFERENTIATION-003 · ★★★ Advanced

If \(x^{3}+y^{3}=6xy\), what is \(\dfrac{dy}{dx}\)?

Choose one option.

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C. \(\frac{x^{2} - 2 y}{2 x - y^{2}}\)

Explanation

Differentiate both sides with respect to x, treating y as a function of x, and solve for \(\dfrac{dy}{dx}\): the result is \(\frac{x^{2} - 2 y}{2 x - y^{2}}\).

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Review the concept: Implicit differentiation

When \(y\) is defined implicitly by an equation in \(x\) and \(y\), both sides are differentiated with respect to \(x\), remembering that \(y\) depends on \(x\) (chain rule), and then \(\frac{dy}{dx}\) is solved for.

\[x^{2}+y^{2}=25\ \Rightarrow\ 2x+2y\,y'=0\ \Rightarrow\ y'=-\frac{x}{y}\]

If \(x^{3}+y^{3}=6xy\), what is \(\dfrac{dy}{dx}\)?

Explanation

Review the concept: Implicit differentiation

More practice problems: implicit differentiation