Use implicit differentiation to find dy/dx for x^3 + x^2*y + y^3 = 7.

Step-by-step solution

1. 1

   Decide why implicit differentiation is needed. The relation cannot be solved cleanly for $y$, so instead differentiate both sides with respect to $x$ while treating $y$ as an unknown function $y(x)$. Every appearance of $y$ then produces a factor $\frac{dy}{dx}$ through the chain rule.

2. 2

   Differentiate the mixed term with the product rule. The term $x^{2}y$ is a product of two functions of $x$, so $\frac{d}{dx}\left(x^{2}y\right)=x^{2}\frac{dy}{dx}+2xy .$ Forgetting the $2xy$ half here is the most common error in this problem.

3. 3

   Differentiate the remaining terms. The pure power gives $\frac{d}{dx}(x^{3})=3x^{2}$; the chain rule gives $\frac{d}{dx}(y^{3})=3y^{2}\frac{dy}{dx}$; and the constant gives $\frac{d}{dx}(7)=0$. Putting everything together, $3x^{2}+x^{2}\frac{dy}{dx}+2xy+3y^{2}\frac{dy}{dx}=0 .$

4. 4

   Collect the dy/dx terms on one side. Grouping, $\left(x^{2}+3y^{2}\right)\frac{dy}{dx}=-3x^{2}-2xy .$

5. 5

   Solve and note where the formula fails. Dividing gives $\frac{dy}{dx}=-\frac{3x^{2}+2xy}{x^{2}+3y^{2}} .$ This is valid wherever $x^{2}+3y^{2}\ne 0$, i.e. everywhere except the origin, which does not lie on the curve since $0\ne 7$; so the formula holds at every point of the curve.