Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

22. \({{\rm{x}}^{{\raise0.7ex\hbox{${\rm{2}}$} \!\mathord{\left/{\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ + }}{{\rm{y}}^{{\raise0.7ex\hbox{${\rm{2}}$}\!\mathord{\left/{\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ = 4, }}\left( {{\rm{ - 3}}\sqrt {\rm{3}}{\rm{,1}}} \right)\) (astroid)

The given equation is \({{\rm{x}}^{{\raise0.7ex\hbox{${\rm{2}}$} \!\mathord{\left/{\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ + }}{{\rm{y}}^{{\raise0.7ex\hbox{${\rm{2}}$}\!\mathord{\left/{\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ = 4}}\) with the point \(\left( {{\rm{ -3}}\sqrt {\rm{3}} {\rm{,1}}} \right)\)

Implicit differentiation: This method consists of differentiating both sides of the equation with respect to \({\rm{x}}\) and then solving the resulting equation for \({\rm{y'}}\)

The equation is

\({{\rm{x}}^{{\raise0.7ex\hbox{${\rm{2}}$} \!\mathord{\left/{\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ + }} {{\rm{y}}^{{\raise0.7ex\hbox{${\rm{2}}$}\!\mathord{\left/{\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ = 4,}}\)

Differentiate both sides with respect to \({\rm{x}}\)

\(\frac{{\rm{d}}}{{{\rm{dx}}}}\left( {{{\rm{x}}^{{\raise0.7ex\hbox{${\rm{2}}$} \!\mathord{\left/ {\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ + }}{{\rm{y}}^{{\raise0.7ex\hbox{${\rm{2}}$} \!\mathord{\left/{\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}} \right){\rm{ = }}\frac{{\rm{d}}}{{{\rm{dx}}}}\left( {\rm{4}} \right)\)

The Sum rule for differentiation

\(\frac{{\rm{d}}}{{{\rm{dx}}}}\left( {{\rm{f}}\left( {\rm{x}} \right){\rm{ + g}}\left( {\rm{x}} \right)} \right){\rm{ = }}\frac{{{\rm{d}}\left( {{\rm{f}}\left( {\rm{x}} \right)} \right)}}{{{\rm{dx}}}}{\rm{ + }}\frac{{{\rm{d}}\left( {{\rm{g}}\left( {\rm{x}} \right)} \right)}}{{{\rm{dx}}}}\)

\(\frac{{\rm{d}}}{{{\rm{dx}}}}\left( {{{\rm{x}}^{{\raise0.7ex\hbox{${\rm{2}}$} !\mathord{\left/{\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}} \right){\rm{ + }}\frac{{\rm{d}}}{{{\rm{dx}}}}\left({{{\rm{y}}^{{\raise0.7ex\hbox{${\rm{2}}$} \!\mathord{\left/ {\vphantom {{\rm{2}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}} \right){\rm{ = 0}}\)

Chain rule: The chain rule states that the derivative of

\({\rm{f}}\left( {{\rm{g}}\left( {\rm{x}} \right)} \right)\) is equal to \({\rm{f'}}\left({{\rm{g}}\left( {\rm{x}} \right)} \right){\rm{ \times g'}}\left( {\rm{x}} \right)\)

\(\begin{aligned}\frac{2}{3}{{\rm{x}}^{{\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/{\vphantom {{ 1}{\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ + }}\frac{2}{3}{{\rm{y}}^{{\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/{\vphantom {{ - 1}{\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{y' = 0}}\\{{\rm{x}}^{{\raise0.7ex\hbox{${ -1}$} \!\mathord{\left/{\vphantom {{ - 1} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ +}}{{\rm{y}}^{{\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/{\vphantom {{ - 1}{\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{y' = 0}}\end{aligned}\)

Take \({\rm{y'}}\) common from the two terms in left hand side

\(\begin{aligned}{{\rm{y}}^{{\raise0.7ex\hbox{${{\rm{ - 1}}}$} \!\mathord{\left/ {\vphantom {{{\rm{ -1}}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}y' &= - {{\rm{x}}^{{\raise0.7ex\hbox{${{\rm{ - 1}}}$}\!\mathord{\left/ {\vphantom {{{\rm{ - 1}}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}\\y' &= \frac{{{\rm{ - }}{{\rm{x}}^{{\raise0.7ex\hbox{${{\rm{ - 1}}}$} \!\mathord{\left/ {\vphantom {{{\rm{ - 1}}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}}}{{{{\rm{y}}^{{\raise0.7ex\hbox{${{\rm{ - 1}}}$} \!\mathord{\left/ {\vphantom {{{\rm{ - 1}}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}}}& = -{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)^{{\raise0.7ex\hbox{${{\rm{ - 1}}}$} \!\mathord{\left/{\vphantom {{{\rm{ - 1}}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}&= - {\left( {\frac{{\rm{y}}}{{\rm{x}}}} \right)^{{\raise0.7ex\hbox{${\rm{1}}$} \!\mathord{\left/ {\vphantom {{\rm{1}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}\end{aligned}\)

The tangent line of a curve at a given point is a line that just touches the curve at that point and the slope of the tangent line of \({\rm{y = f}}\left( {\rm{x}} \right)\) at a point \(\left( {{{\rm{x}}_{\rm{0}}}{\rm{,}}{{\rm{y}}_{\rm{0}}}} \right)\) is \({\left. {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right|_{\left( {{{\rm{x}}_{\rm{0}}}{\rm{,}}{{\rm{y}}_{\rm{0}}}} \right)}}\)

So the slope of the tangent line at \(\left( {{\rm{ - 3}}\sqrt {\rm{3}} {\rm{,1}}} \right)\) is

\({\left. {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right|_{\left( {{\rm{ - 3}}\sqrt {\rm{3}} {\rm{,1}}} \right)}}{\rm{ = - }}{\left( {\frac{{\rm{1}}}{{{\rm{ - 3}}\sqrt {\rm{3}} }}} \right)^{{\raise0.7ex\hbox{${\rm{1}}$} \!\mathord{\left/{\vphantom {{\rm{1}} {\rm{3}}}}\right.}\!\lower0.7ex\hbox{${\rm{3}}$}}}}{\rm{ = }}\frac{{\rm{1}}}{{\sqrt {\rm{3}} }}\)

The tangent line formula is,

\({\rm{y - }}{{\rm{y}}_{\rm{0}}}{\rm{ = }}{\left. {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right|_{\left( {{{\rm{x}}_{\rm{0}}}{\rm{,}}{{\rm{y}}_{\rm{0}}}} \right)}}\left( {{\rm{x - }}{{\rm{x}}_{\rm{0}}}} \right)\)

An equation of the tangent line at \(\left( {{\rm{ - 3}}\sqrt {\rm{3}} {\rm{,1}}} \right)\) is

\(\begin{aligned}y - 1 &= \frac{{\rm{1}}}{{\sqrt {\rm{3}} }}\left( {{\rm{x - }}\left( {{\rm{ - 3}}\sqrt {\rm{3}} } \right)} \right)\\y - 1 &=\frac{{\rm{1}}}{{\sqrt {\rm{3}} }}{\rm{x + 3}}\\y &= \frac{{\rm{1}}}{{\sqrt {\rm{3}} }}{\rm{x + 4}}\end{aligned}\)

Therefore, the equation of the tangent line at \(\left( {{\rm{ - 3}}\sqrt {\rm{3}} {\rm{,1}}} \right)\)is \({\rm{y = }}\frac{{\rm{1}}}{{\sqrt {\rm{3}} }}{\rm{x + 4}}\)