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Implicit differentiation

If we want to differentiate an implicit function such as \( y^2 + y =
x + x^3 \), we notice that by the chain rule,

\begin{displaymath}
\frac{\mathrm{d}}{\mathrm{d}x} f(y) = f'(y)
\frac{\mathrm{d}y}{\mathrm{d}x}
\end{displaymath}

We can use this results to differentiate the above function as follows.

\begin{eqnarray*}
y^2 + y & = & x + x^3 \\
2y \frac{\mathrm{d}y}{\mathrm{d}x} +...
...
\frac{\mathrm{d}y}{\mathrm{d}x} & = & \frac{1 + 3 x^2}{2y + 1}
\end{eqnarray*}





Alexander Frolkin 2001-03-13