Question

Find \(d y / d x\) by implicit differentiation. $$ x^{2} y+y^{2} x=-3 $$

Short answer

The derivative of the function is given by \(dy/dx = (-2xy - y^{2}) / (x^{2} + 2xy)\)

Step-by-step solution

Differentiate each term using the chain rule

The function can be considered as three separate terms for the purpose of differentiation: \(x^{2} y\), \(y^{2} x\), and \(-3\). The derivative of each term will be found by applying the product rule and chain rule where necessary: 1) \(d/dx(x^{2} y) = 2x*y + x^{2}*(dy/dx)\) We applied the product rule here: \(d/dx(uv) = v*du/dx + u*dv/dx\), where \(u=x^{2}\) and \(v=y\) 2) \(d/dx(y^{2} x) = y^{2} + 2xy*(dy/dx)\) Again the product rule was applied here, but this time \(u=y^{2}\) and \(v=x\) 3) \(d/dx(-3) = 0\) because the derivative of a constant is zero.

Substitute the derivatives for their respective terms in the original function

Once each term has been differentiated, they can be replaced in the function: So, the derivative of the function \(x^{2} y + y^{2} x = -3\) is: \(2x*y + x^{2}*(dy/dx) + y^{2} + 2xy*(dy/dx) = 0\)

Solve for \(dy/dx\)

Rearrange the equation to derive \(dy/dx\). Group the terms with \(dy/dx\) on one side and the remaining terms on the other side of the equation: \(x^{2}*(dy/dx) + 2xy*(dy/dx) = -2xy - y^{2} \), Group the terms with \(dy/dx\): \(dy/dx * (x^{2} + 2xy) = -2xy - y^{2}\), Finally solve for \(dy/dx\): \(dy/dx = (-2xy - y^{2}) / (x^{2} + 2xy)\)