Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 + 2xy − y2 + x = 20, (3, 4) (hyperbola)

Answer 1

$\bf x^2+2x6-y^2+x=20\impliedby taking\ \cfrac{dy}{dx}\textit{ to both sides} \\\\\\ 2x+2\left( y+x\frac{dy}{dx}\right)-2y\frac{dy}{dx}=0\impliedby \textit{common factor 2} \\\\\\ 2\left[ x+\left( y+x\frac{dy}{dx}\right)-y\frac{dy}{dx}\right]=0 \\\\\\ x+y+x\frac{dy}{dx}-y\frac{dy}{dx}=0\impliedby \textit{common factor }\frac{dy}{dx} \\\\\\ \cfrac{dy}{dx}(x-y)=-x-y\implies \cfrac{dy}{dx}=\cfrac{-x-y}{x-y}\qquad \begin{cases} x=3\\ y=4 \end{cases}\ then$

$\bf \cfrac{dy}{dx}=\cfrac{-3-4}{3-4}\implies \cfrac{dy}{dx}=7\\\\ -----------------------------\\\\ \textit{tangent equation at }\begin{cases} x=3\\ y=4\\ m=7 \end{cases} \\\\\\ y-4=7(x-3)\implies y=7x-21+4\implies y=7x-17$