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Reciprocal rule

Since by the chain rule,

\begin{eqnarray*}
\frac{\mathrm{d}y}{\mathrm{d}x} \frac{\mathrm{d}x}{\mathrm{d}y} & = &
\frac{\mathrm{d}y}{\mathrm{d}y} \\
& = & 1
\end{eqnarray*}



We can rearrange this to find

\begin{displaymath}
\frac{\mathrm{d}y}{\mathrm{d}x} =
\frac{1}{\frac{\mathrm{d}x}{\mathrm{d}y}}
\end{displaymath}

This result is useful when finding the derivative of a function when the derivative of its inverse function is known. For example, to find \( \frac{\mathrm{d}}{\mathrm{d}x} \ln{x} \),

\begin{eqnarray*}
y & = & \ln{x} \\
x & = & \exp{y} \\
\frac{\mathrm{d}x}{\mat...
...{d}y}{\mathrm{d}x} & = & \frac{1}{\exp{y}} \\
& = & \frac{1}{x}
\end{eqnarray*}





Alexander Frolkin 2001-03-13