Answer
The tangent line to the curve at $P(1,1)$ is $$y=\frac{5}{13}x+\frac{8}{13}$$
Work Step by Step
$$2y^2+(xy)^{1/3}=x^2+2$$
The graph of the curve is enclosed below.
a) As the point $(1,1)$ lies in the curve, $P$ satisfies the equation.
b) Find $dy/dx$ using implicit differentiation: $$4y\frac{dy}{dx}+\frac{1}{3}(xy)^{-2/3}\frac{d}{dx}(xy)=2x$$ $$4y\frac{dy}{dx}+\frac{1}{3}(xy)^{-2/3}(y+x\frac{dy}{dx})=2x$$ $$4y\frac{dy}{dx}+\frac{1}{3}(xy)^{-2/3}y+\frac{1}{3}(xy)^{-2/3}x\frac{dy}{dx}=2x$$ $$\frac{dy}{dx}\Big(4y+\frac{1}{3}(xy)^{-2/3}\Big)=2x-\frac{1}{3}(xy)^{-2/3}y$$ $$\frac{dy}{dx}=\frac{2x-\frac{1}{3}(xy)^{-2/3}y}{4y+\frac{1}{3}(xy)^{-2/3}}$$
- For $P(1,1)$: $$\frac{dy}{dx}=\frac{2\times1-\frac{1}{3}(1\times1)^{-2/3}\times1}{4\times1+\frac{1}{3}(1\times1)^{-2/3}}=\frac{2-\frac{1}{3}\times1}{4+\frac{1}{3}\times1}=\frac{\frac{5}{3}}{\frac{13}{3}}=\frac{5}{13}$$
c) The tangent line to the curve at $P(1,1)$ is $$y-1=\frac{5}{13}(x-1)$$ $$y-1=\frac{5}{13}x-\frac{5}{13}$$ $$y=\frac{5}{13}x+\frac{8}{13}$$
