Differentiation of Implicit Functions Examples

Example: Find \frac{{dy}}{{dx}}, if the given implicit function is

{x^3} + {y^3} = xy

We have the given implicit function

{x^3}  + {y^3} = xy

Differentiate with respect to x, we have

\frac{d}{{dx}}{x^3}  + \frac{d}{{dx}}{y^3} = \frac{d}{{dx}}\left( {xy} \right)


Where xy is product of two variable and using product of derivative, we have

\begin{gathered} 3{x^2} + 3{y^2}\frac{{dy}}{{dx}} =  x\frac{{dy}}{{dx}} + y\frac{d}{{dx}}x \\ \Rightarrow 3{x^2} + 3{y^2}\frac{{dy}}{{dx}}  = x\frac{{dy}}{{dx}} + y \\ \Rightarrow 3{y^2}\frac{{dy}}{{dx}} -  x\frac{{dy}}{{dx}} = y - 3{x^2} \\ \Rightarrow \left( {3{y^2} - x}  \right)\frac{{dy}}{{dx}} = y - 3{x^2} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{y - 3{x^2}}}{{3{y^2} - x}} \\ \end{gathered}

Example: Find the derivative of implicit function

\sqrt  x - \sqrt y = \sqrt {xy}

We have the given implicit function

\sqrt  x - \sqrt y = \sqrt {xy}

Differentiate with respect to x, we have

\begin{gathered} \frac{d}{{dx}}\sqrt x - \frac{d}{{dx}}\sqrt y = \frac{d}{{dx}}\sqrt {xy} \\ \Rightarrow \frac{1}{{2\sqrt x }} -  \frac{1}{{2\sqrt y }}\frac{{dy}}{{dx}} = \frac{1}{{2\sqrt {xy}  }}\frac{d}{{dx}}\left( {xy} \right) \\ \end{gathered}


Where xy is product of two variable and using product of derivative, we have

\begin{gathered} \frac{1}{{2\sqrt x }} - \frac{1}{{2\sqrt y  }}\frac{{dy}}{{dx}} = \frac{1}{{2\sqrt {xy} }}\left( {x\frac{{dy}}{{dx}} +  y\frac{d}{{dx}}x} \right) \\ \Rightarrow \frac{1}{{2\sqrt x }} -  \frac{1}{{2\sqrt y }}\frac{{dy}}{{dx}} = \frac{1}{{2\sqrt {xy} }}\left(  {x\frac{{dy}}{{dx}} + y} \right) \\ \Rightarrow \frac{1}{{2\sqrt x }} - \frac{1}{{2\sqrt y }}\frac{{dy}}{{dx}} = \frac{x}{{2\sqrt {xy}  }}\frac{{dy}}{{dx}} + \frac{y}{{2\sqrt {xy} }} \\ \Rightarrow \frac{1}{{2\sqrt x }} - \frac{1}{{2\sqrt y }}\frac{{dy}}{{dx}} = \frac{{\sqrt x }}{{2\sqrt y  }}\frac{{dy}}{{dx}} + \frac{{\sqrt y }}{{2\sqrt x }} \\ \Rightarrow \frac{{\sqrt x }}{{2\sqrt y  }}\frac{{dy}}{{dx}} + \frac{1}{{2\sqrt y }}\frac{{dy}}{{dx}} = \frac{1}{{2\sqrt  x }} - \frac{{\sqrt y }}{{2\sqrt x }} \\ \Rightarrow \left( {\frac{{\sqrt x  }}{{2\sqrt y }} + \frac{1}{{2\sqrt y }}} \right)\frac{{dy}}{{dx}} =  \frac{1}{{2\sqrt x }} - \frac{{\sqrt y }}{{2\sqrt x }} \\ \Rightarrow \left( {\frac{{\sqrt x + 1}}{{2\sqrt y }}} \right)\frac{{dy}}{{dx}}  = \frac{{1 - \sqrt y }}{{2\sqrt x }} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{1 - \sqrt y }}{{2\sqrt x }} \times \frac{{2\sqrt y }}{{\sqrt x + 1}} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{\sqrt  y - y}}{{x + \sqrt x }} \\ \end{gathered}