Given that ${{x}^{3}}-x\sin \left( y \right)=3xy$, find an expression of \(\frac{\mathrm{d}y}

Question

Given that ${{x}^{3}}-x\sin \left( y \right)=3xy$, find an expression of $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Accepted Answer

For $x\ne0$, $\frac{\mathrm dy}{\mathrm dx}=\frac{3x^2-\sin y-3y}{x(3+\cos y)}=\frac{2x}{3+\cos y}$. The relation also contains the vertical component $x=0$; its selected smooth nonvertical branch has slope $0$ at $(0,0)$.

Techniques of Differentiation - Implicit Differentiation

Techniques of Differentiation - Implicit Differentiation

Solution:

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